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\NoRunningHeads
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\def\er{\Bbb R}
\def\en{\Bbb N}
\def\zet{\Bbb Z}
\def\de{\Bbb D}
\def\pe{\Bbb P}
\def\es{\Bbb S}
\def\ex{\Bbb X}
\def\ee{\Bbb E}
\def\Af{A_{\text{full}}^{*}}
\def\Am{A_{\text{max}}^{*} }
\def\Gammab{\boldsymbol\Gamma}
\def\gb{{\boldsymbol\Gamma}}
\def\vv{{V_{\updownarrow}(\gb)}}
\def\vvv{{V_{\updownarrow}(\gb')}}
\def\vvi{{V_{\updownarrow}(\gb_i)}}
\def\vvd{{V_{\updownarrow}(\Cal D)}}
\def\vva{{V_{\updownarrow}(A)}}
\def\vvb{{V_{\updownarrow}(B)}}
\def\vvc{{V_{\updownarrow}(\tilde B)}}
\def\vvt{V_{\updownarrow}(T)}
\def\vvg{V_{\updownarrow}(G)}
\def\ccap{\cap \cap}
\def\zv{\Bbb Z^{\nu}}
\def\zw{\Bbb Z^{\nu-1}}
\def\ps{Pirogov -- Sinai}{}
\def\card{|operatorname{card}}
\def\diam{\operatorname{diam}}
\def\dist{\operatorname{dist}}
\def\supp{\operatorname{supp}}
\def\card{\operatorname{card}}
\def\conn{\operatorname{conn}}
\def\con{\operatorname{con}}
\def\Conn{\operatorname{Conn}}
\def\ssubset{\subset \subset}
\def\ext{\operatorname{ext}}
\def\loc{\operatorname{loc}}
\def\full{\operatorname{full}}
\def\sign{\operatorname{sign}}
\def\obr#1{\vskip2cm\centerline{\bf}\vskip2cm}
\def\df{\flushpar{\bf Definition. }}
\topmatter
\title Stratified Low Temperature Phases of Stratified
Spin Models : A General Pirogov -- Sinai Approach
\endtitle
\author Petr Holick\'y and Milo\v s Zahradn\'\i{}k
\endauthor
\affil Faculty of Mathematics and Physics, Charles University,
Sokolovsk\'a 83, 186 00 Prague, Czech Republic
\endaffil
\email holicky\@karlin.mff.cuni.cz, mzahrad\@karlin.mff.cuni.cz
\endemail
\thanks Partially supported by: Commission of the European Union
under contracts CHRX-CT93-0411 and CIPA-CT92-4016,
Czech Republic grants 202/96/0731
and 96/272.
\endthanks
\date August 30, 1997
\enddate
\keywords Low temperature Gibbs states, stratified phases
(with many interfaces) for stratified and nonstratified
Hamiltonians, ``local'' ground states, interfaces,
contours, Dobrushin's walls, Pirogov -- Sinai theory,
Peierls condition, contour functional,
``metastable'' submodels, (inductively organized)
cluster expansions, ground states of one dimensional models,
phase diagrams
\endkeywords
\subjclass 82A25
\endsubjclass
\abstract We adapt and improve the existing Pirogov -- Sinai technology
to
obtain a general and unifying approach to the study of low
temperature ``stratified''
phases for classical spin models whose Hamiltonian may not even be
translation invariant but is ``stratified'', i.e.
invariant with respect to all ``horizontal'' shifts
(not changing the last coordinate). Examples are
``stratified'' versions of classical models like the Ising model with
``vertically dependent'' external field; models in halfspaces or layers
and also those translation invariant models where
Dobrushin's
phases with
rigid interfaces (one or more) appear. Our method brings some
clarification and sharpening even when applied to the ordinary
situations of the Pirogov -- Sinai theory.
Our main result transcripts the question of characterizing the
``stratified''
Gibbs states of the given model to the question of finding the
{\it ground states} of some auxiliary {\it one dimensional\/} model with
infinite range but quickly decaying interactions.
\endabstract
\endtopmatter
\document
\head I. Introduction, Notes on the Development of the Problem and
Some Examples
\endhead
The rigorous study of Gibbs states with translation noninvariant
structure exhibiting a ``rigid interface
between the translation invariant phases above and below
the interface'',
goes back to the pioneering
Dobrushin's paper \cite{1}.
Several authors continued this study; we note e.g\. article
\cite{HKZ} where an attempt to combine basic Dobrushin's
ideas with the power of Pirogov -- Sinai theory was made.
The leading idea in these investigations was to transcript the problem
of description of the structure of the rigid interface to a suitable
{\it lower dimensional\/} problem.
In more concrete terms, using the expansion of the partition sums
above and below the interface, the behaviour of the ``walls''
of the interface between the $+$ and $-$ phases in the three dimensional
Ising model can be viewed as a behaviour of contours of some auxiliary
two dimensional perturbed Ising model.
In \cite{HZ}, we applied a similar approach based on the reduction to a
lower dimensional problem to the study of wetting phenomena and entropic
repulsion in the Ising model in halfspace.
During our attempt to pursue the method to other interesting
situations like the study of ``wetting layers'' emerging in some phases of
the Blume -- Capel model and also in the order -- disorder --
(other) order
phases appering in the Potts model below the critical temperature
(the article \cite{HMZ} is under preparation),
we found that the additional technical problems are forcing us to look
for a more appropriate method. Finally we were lead to a conclusion
that the ``dimensional reduction method'' based on this particular kind
of a partial exansion of the considered model should be abandoned.
Instead, we found a modification of the Pirogov --
Sinai theory which applies
{\it directly\/}
to these ``stratified'' situations. We hope that the fact that our new
version of the Pirogov -- Sinai theory gives even some
new insight and a
simplification into
the traditional ``translation invariant'' Pirogov -- Sinai theory confirms that the
method developed by us is adequate.
Methodologically, our approach
is based on the version \cite{Z} of the Pirogov -- Sinai theory but the concept of a
``stable'' (``small'') contour and of a ``metastable ensemble''
is now investigated in a greater detail.
Moreover, the concept of a contour ensemble now {\it disappeared\/}
from our version of Pirogov -- Sinai theory completely!
The concept of a suitably defined ``contour
functional'' $F(\gb)$ (as compared to
the contour energy $E(\gb)$) remains in our approach
as an extremely important
{\it testing quantity\/} -- allowing one to decide whether the contour
is ``well behaved enough''--
but instead of the construction of auxiliary contour
models, the central point of our approach is now an idea of a
successive partial {\it expansion\/} of the model. This is based
on an important new
technical step which is called {\it recoloring\/} of a contour here.
Recoloring of a contour $\gb$ in a partially expanded model means that
a new, ``more expanded'' model with the {\it same\/} partition functions
is constructed where $\gb$ does not appear anymore.
We will see that
the ``metastable'' submodels of the given model
(constructed for any stratified boundary
condition) can be {\it expanded completely\/}
and moreover for the ``stable'' boundary conditions, the corresponding
metastable model will
be identical with the original ``physical'' model.
The organization
of our expansions will make unnecessary previous
estimates like
``Main Lemma'' used in \cite{Z}.
Instead, we have now a more powerful method based on our Theorems 5
and 6.
To summarize, we converted the Pirogov -- Sinai theory just to a carefully
organized {\it method\/} of (successive)
{\it expansion of suitable partition functions\/}.
The use of expansion techniques is absolutely crucial
in our situation and the construction of the expansions
is a more delicate task than in the translation invariant
situations studied before. Namely, contours of the models studied so far
were ``crusted'' in the sense that the events outside and inside the
contour were independent. This is {\it not\/} valid
here in our new situation, where contours can be
often interpreted only as
``walls'' (the terminology of \cite{D}) of the interface(s)
and then the events
happening ``inside'' resp\. ``outside'' of the wall
are {\it not\/} independent ones.
This problem was solved in \cite{HKZ} by taking expansions
``above'' and ``below'' the
interface and by replacing the walls by more complex ``aggregates''
of walls and clusters
\footnote{ The whole situation was then projected to
$\zet^{\nu -1}$ which is the main idea of the paper \cite{D}.}
but for complicated phases with {\it several\/}
interfaces such an approach is too complicated.
Now we treat both the ``crusted'' contours and the ``noncrusted''
ones (walls) in the {\it same way\/}. However,
the fact that some contours
are not ``crusted'' implies that their
contour functional $F(\gb)$
(see below for an extensive discussion of this
``testing quantity'')
must be now
defined
much more carefully \footnote{Retaining its meaning, vaguely speaking,
of the ``work needed to install the given contour''.}.
We construct succesively, by induction, the expansion
of the {\it whole metastable model\/}, leaving out
the previous idea (of \cite{HKZ}) of the expansion in {\it two\/}
different steps
(first the expansion of the ensemble of contours and then of the
ensemble
of the walls resp. of the ``aggregates'', see \cite{HKZ}).
\newline
Let us now mention some typical examples which can be treated
by our method -- which often gives
results more
powerful than those obtained by previous methods.
Besides of the fact that we expect a {\it general
applicability\/} of our method to all of these examples, one
should stress also that for each special model,
the study of the ``important'' terms of the
cluster expansion series must be done relevantly
to the needs of the particular model
to obtain useful results \footnote{For example,
to establish the ``
logarithmic'' width
of the middle layer in the model (2 a,b) below.}
and each of the models mentioned below surely requires
a detailed treatment.
\bigskip
(1 a) Models in halfspace
$\zv_+=\{t\in\zv;t_{\nu}\geq 0\}$
with ``unstable'' boundary condition on the bottom (like the $-$
boundary condition for the ferromagnetic Ising model with a negative external
field, making $+$ the only ground state of the model).
Than the ``Basuev states''
(terminology of R\. L\. Dobrushin)
with a wetting layer of minuses appear.
(1 b) Models in layers (like in \cite{MS}, \cite{MDS}).
\smallskip
(2 a) Models of Blume -- Capel type with spins belonging to some
finite set $Q\subset \er$ and with the Hamiltonian consisting of
a (say) quadratic pair interaction and a potential $V$:
$$
H(x) =\sum_{(t,s)}(x_t-x_s)^2+\sum_t V(x_t) \tag 1.1
$$
where $V$ has several ``potential wells'' of
approximately the same depth. Consider the case of
{\it three\/} wells, for example.
If $x_+$, $x_0$, $x_-$ mark the
bottoms of three adjancent wells of $V$ then it may
happen, for suitable choice of $V$, that both $x_+$ and
$x_-$ give rise to a stable phase while $x_0$ is unstable.
Then one should expect also the existence of a phase which
``goes vertically from $x_+$ to $x_-$ through a layer of a
metastable $0$--th phase''. The question is about the
determination of the width of the $0$--th layer.
On a more concrete level, take the ordinary
Blume -- Capel model on $\{-1,0,+1\}^{\zv}$ with a very strong
repulsion between $+$ and $-$ and with a weaker, but still strong
repulsion between $\pm1$ and $0$ (see \cite{BS} or
\cite{ZR} for more information)
with the boundary condition $\sign x_t$
being prescribed outside of some
finite volume $\Lambda$ and only {\it partially\/},
for $|t_{\nu}| > n $ where
$n$ is sufficiently large.
The question is what picture is obtained if
$\Lambda \to \zv$ , for any extension of the
boundary conditions above.
Somehow more comfortable variant
of such a question
is the following one:
what happens if we fix the above boundary conditions
$+1$ resp. $-1$ for $t_{\nu} \geq n$ resp.
$t_{\nu} \leq -n$
and in the middle layer with $|t_{\nu}| \leq n-1$,
$ n$ being a suitable
integer, we take the spin value zero ?
{\it This\/} is the very kind of a question we pose in this
paper. The precise value of $\tilde n$ will be known
in general only at the very
{\it end\/} of our investigation !
Our main theorem says, roughly speaking, that it is possible to
prescribe boundary condition for these smaller $|t_{\nu}| $
such that a {\it rigid\/} interface formed by {\it two\/} paralell
(slightly perturbed somewhere) planes -- with the
layer of (mostly) zero valued spins inside -- appears for
{\it any\/} reasonable (finite or infinite) volume $\Lambda$.
Of course the subtle question of the exact thickness of
such a ``zero phase layer'' inside
can be solved only when looking at the quantities
$\{h_t\}$ in more detail.
This is now a {\it specialized\/} question
which can be studied
(after the general technology of the computation
of these quantities, and {\it this\/}
is the essence of the presented paper, was shown to work)
for any particular model.
Our result assures
that the series defining the quantities
$\{h_t\}$ very quickly converge and therefore
one can give a prescription on precisely how many
``first few terms'' of the expansions have to be computed
in order to select, from the competing candidates for the
thickness
of the layer, the ``best'' one minimizing the
free energy of the whole interface.
\footnote{ To say this once again: our theorem
guarantees the {\it existence\/} of the
rigid interface in the infinite volume
Gibbs state; however the computation of the precise width of the
layer is a more specialized question.
A comparison of the competing terms in the expansion series
is developed in detail
in the prepared article \cite{MZ} -- which deals with
the above mentioned Blume -- Capel model and also with the Potts
model.}
(2 b) A rigid interface with a thick middle layer
appears also
for the Potts model with a large number of spins
slightly below the critical temperature, where phases of
the type \newline ``one
order above $|$ layer of disorder $|$ another order below''
exist.
Using the Fortuin -- Kasteleyn
representation this situation can be described analogously
as above (see \cite{HMZ}).
\smallskip
(3 a) ``Sedimentary Ising rock''. ``Stratified \ps\
models''.
Consider some ordinary
translation invariant Pirogov -- Sinai
type model and add to it a small
perturbative Hamiltonian which is
invariant with respect to the $\Bbb Z^{\nu-1}$ shifts
(we identify here $\Bbb Z^{\nu-1}$ with the subspace
$\Bbb Z^{\nu-1} \times \{0\}$ of $\zv$)
i.e\. depends on the last,``vertical'' coordinate $t_{\nu}$ of
$t=(t_1,\dots,t_{\nu})\in \zv$ only.
Then, one should expect phases with a possibly rich
structure of layers
of ``stable
or slightly instable translation invariant phases
of the unperturbed Hamiltonian''.
For example if one adds, to a ferromagnetic Ising model, a
small, random (this is the most interesting case)
``horizontally invariant'' external
field with approximately zero mean over the vertical
shifts, one should expect phases with {\it
infinitely many\/} $\pm$ layers and the problem is to
compute
the positions of the layers.
(3 b) In fact, the class of ``horizontally
invariant models'' fitting our scheme is much broader and
many examples which are not small perturbations of
translation invariant models can be constructed.
For the illustration of the power of our method
consider the situation of
the paper \cite{EMZ} (under preparation) where the following
situation is studied:
take some {\it one dimensional\/} model (either of the
Ising type or not; preferably a model with
an interesting family of its {\it ground states\/} --
like the
translation invariant model producing nonperiodic Thue Morse sequences
which is considered in \cite{EMZ})
and consider a {\it three\/} (more generally $\nu$--) dimensional
model by adding to the given one dimensional model of
``vertical interactions'' also a sufficiently strong Ising ferrromagnetic
``horizontal interaction'' acting in the directions perpendicular
to the $\nu$ -- th axis.
Our main theorem assures then that the stratified Gibbs states of such
a $\nu$-- dimensional Hamiltonian correspond to the ground states
of some small (small for large temperatures) perturbation
of the original one dimensional Hamiltonian.
In fact in \cite{EMZ} the emphasis is on the {\it opposite\/} question:
on the finding of the
``preimage'' of the given interesting one dimensional Hamiltonian,
i.e. on the finding of a slightly perturbed Hamiltonian having
the property that,
when
``stabilized by strong ferromagnetic horizontal interactions'' it
yields a family of stratified Gibbs states corresponding
exactly to the ground states of the original one dimensional Hamiltonian.
Our main result is formulated in detail in part III, section 8,
and its summary is given already at the end of part II.
Let us outline its meaning:
In the translation invariant Pirogov -- Sinai theory, one
constructs, for any reference translation invariant configuration (for
any translation invariant ``local ground state'')
a quantity
called the ``metastable free energy''. If the minimum of
these quantities
is attained for some configuration $y$, then the Gibbs state characterized as
the
``local perturbation of $y$'' exists (\cite{Z}).
Here, our ``reference configurations'' are (all!) stratified configurations;
instead of
quantities mentioned above we construct some auxiliary
{\it one dimensional\/} model
whose configurations
correspond to various ``horizontally invariant regimes''
of the original model. The various
{\it ground states } of this one dimensional model
correspond to the various {\it stratified Gibbs states\/}
of our original model.
This is our Main Theorem (section 8); the quantities $h_t(y)=h_{t_{\nu}}(y),
\ t
\in \zv $
constructed
there give
all the essential information about the model.
These quantities are, at least in principle, computable. They
are given by cluster expansion series with maybe complicated, but
very quickly decaying terms. In the case
when $y$ is the ground state of the corresponding
one dimensional model (this property will be called
the ``stability of $y$'') the quantities
$h_{t_{\nu}}(y)$ have the physical
interpretation of the ``density of free energy of the $y$--th Gibbs state
at the vertical level $t_{\nu}$'' .
\remark{Concluding Notes} 1.
We are concentrated, in this paper, in the investigation of a phase picture
for a {\it fixed\/} Hamiltonian. If one is interested
in the investigation of the {\it phase diagram\/} of
particular
models (notice that there are in principle infinite parameters
in the models (3) above!) then such investigation
should be based on the study of
the mapping:
$$ \ \ \{ \ \text{ Hamiltonian}
\longmapsto \text{minimizing configurations
of }\{h_{t_{\nu}}(y)\} \ \} \tag 1.2$$
Then one can apply theorems from the differential calculus of finitely
many (in some models (3) infinitely many)
variables, like the
implicit function theorem.
The study of the {\it differentiability properties\/}
of such a diagram requires in fact a
suitable technical modification of the definition
of the contour functional \footnote{and correspondingly
of the quantity
$h_t(y) $.
Such a modification could however act on the nonground values
of
$\{h_t(y)\}$ only;
the ground values of $\{h_t(y)\}$ have nontrivial
physical interpretation and there can be no arbitrariness
in their definition!} $F$ --
to obtain sufficiently nice differentiability (even local analyticity)
properties of (1.2).
2. We expect that the new method
presented in this paper will be
applicable also in other situations (even nonstratified ones)
where ``noncrusted'' contours
appear. Notice also that the method is applicable to situations
where one starts (after suitable preparation of the
given ``physical'' model, see part II)
with some {\it ``mixed model''\/} (see part III, section 2
for the definition of this important concept) instead
of the abstract P.S. model.
This is (among other examples,
like the case of infinite range, quickly
decaying
potentials we briefly mention below) also
the case of models with continuous spins (studied in \cite{DZ}) having
several ``potential wells''. In these models,
the expansion around (positive mass) gaussians (approximating the regimes
of the potential wells)
of the ``restricted ensembles''
(of configurations living in the vicinity
of the potential wells) yields a mixed model of the type
studied here, and then the analysis developed in part III
of this paper could be applied to these models, possibly also for the wells
which are not so ``deep'' such that the previous analysis
(like \cite{DZ}) could be applied to them.
\endremark
\definition{Acknowledgements} The second author (M.Z.)
thanks the Erwin Schr\"odinger Institute for its hospitality
during the time of the autumn 1995 semester
``Gibbs random fields and phase transitions''.
Unfortunately, the organizer of the semestr and our teacher
R.L. Dobrushin could not already come.
We dedicate this paper to his memory.
\enddefinition
\head II. General Description of the Considered Model.
Transcription to an Abstract Pirogov -- Sinai Type Model
\endhead
Given a configuration space
$$
\Bbb X= S^{\zv}\,, \,\,\, \nu\geq 3 ,
\tag 2.1$$
where $S$ is a finite set (of ``spins''), we will consider a general
``horizontally invariant''
(called also ``stratified'' below) Hamiltonian on $X$.
This Hamiltonian $H$ will have (at least
in the basic case studied below) a finite
range $r \in \en$.
\footnote{Our method can be immediately extended to some
more general {\it infinite range\/}
Hamiltonians with sufficiently quickly (exponentially!) decaying
interactions.
However, here it is not a
proper place to formulate
such a generalization. For some more details see
Corollary, end of part II and also the discussion after
Main Theorem, part III.
The paper in preparation \cite{EMZ} will give more
information about this problem.}
In what follows we have to consider a suitable norm on
$\zv$ ; let us fix the usage of the $l_{\infty}$ norm
$|\cdot\|=|\cdot\|_\infty$
\footnote{ In fact, the choice of the norm $l_1$ would be
in a sense nicer, stronger
and more natural in some
parts of the paper, like in the beginning of part III
(if one relates it to our very notion of
commensurability). However, in other reasonings the
$l_{\infty}$ norm is more convenient. For example,
the cubes
introduced in the later sections of part III
should be
otherwise replaced by octagons to get the
strongest possible estimates there. We do not want
to do that in this paper.} everywhere.
Put
$$
H(x_{\Lambda }|x_{\Lambda^c })=
\sum
\Sb A\cap\Lambda \ne\emptyset\\
\diam A \leq r
\endSb
\Phi _A(x_A) ,
\tag 2.2
$$
where $\Phi_A$ are some ``interactions'', i.e. functions
on $S^{A}$ with values in $\er \cup +\infty$ which are``stratified'' in the
sense explained below.
We will study the structure of (stratified, see below)
{\it Gibbs states\/}
of the model, more precisely of the probabilities
\footnote{The probabilities
$ P_{\Lambda,\beta} ^{x_{\Lambda ^c}} (\cdot)$ are called
finite volume Gibbs states under boundary condition
$x_{\Lambda ^c}$. } which are
given in finite volumes
$\Lambda $ by formulas
$$
P_{\Lambda,\beta}^{x_{\Lambda ^c}} (x_\Lambda )=
(Z_{\beta}(\Lambda ,x_{\Lambda ^c}))^{-1}
\exp(-\beta H(x_{\Lambda }|x_{\Lambda ^c}))
\tag 2.3
$$
where $\beta =\frac{1}{T}$ is the inverse
temperature. The {\it partition function\/}
$ Z_{\beta}(\Lambda ,x_{\Lambda ^c})$, under boundary condition
$x_{\Lambda^c} \in S^{\Lambda^c}$ is defined by
$$
Z_{\beta}(\Lambda, x_{\Lambda ^c})=
\sum_{x_{\Lambda }}
\exp(-\beta H(x_{\Lambda }|x_{\Lambda ^c})) .
\tag 2.4$$
Suitable {\it infinite volume limits\/}
of these finite volume Gibbs states will be studied below.
\subhead Notes
\endsubhead
{\bf 1.}
In fact, some other (more special than (2.4)) partition functions,
namely
so called (strictly) {\it diluted\/}
partition functions will be
important later. The recurrent structure of the measures (2.3)
-- which is formulated by the DLR equations --
and the concept of a general (nondiluted) partition function under
arbitrary boundary condition will not play, in our later approach,
such an important role as usually.
More adequate for our later approach is the idea that
$\Lambda$ is some (typically very large) volume which will be
{\it fixed\/} in the main part of our future considerations, together
with some very special (``stratified'', see below)
boundary conditions given on its complement.
Only at the very end of the paper
-- when proving and interpreting our Main Theorem, section III.8 --
our ``playground'' $\Lambda$ will be expanded to the
whole $\zv$ and the limit Gibbs states thus obtained will
be investigated.\newline
{\bf 2.} In the following we will {\it always put\/}
$\beta=1$\ i.e. \ we
will include the term $\beta = 1/T$ into the definition of
$\Phi _{A}$ and $H$. Thus the temperature
\footnote{{\it Low\/} temperature in most applications;
see the Peierls condition formulated below in (2.14).}
will be just one of the parameters in the Hamiltonian. We
emphasize that in this paper we are interested only in the
clarification of the phase picture for a given {\it fixed\/}
Hamiltonian. Doing this, one can study the {\it change\/}
of this picture (and of relevant quantities like the free
energies) when some parameters are changing. Our approach
gives some basic tools for doing that: namely
we define useful quantities called {\it metastable free
energies\/} which really govern the behaviour of the phase
diagram -- see our Main Theorem. However, relevant result on
the behaviour of the phase {\it diagram\/} is, in a sense,
not even
formulated in our paper. (See Corollary
at the end of section II for some information.)
\newline
{\bf 3.} One could be interested in the structure of Gibbs
states, under suitable boundary conditions, also for other
infinite volumes like the halfspace $\zv_+=\{t\in\zv;t_{\nu}\geq 0\}$
or in a layer. It is not hard to see that such a situation
could be modeled on $\Lambda =\zv$ by choosing a suitable
modification of the Hamiltonian: for example if we put
$$
\Phi _{A}(x_{A})=+\infty
$$
whenever $A\not\subset\zv_+$ and
$x_{A\cap \zv_{-}}\ne\{x_t=\bar x_{t}, t\in A \cap \zv_{-}\}$
we obtain a limit Gibbs state on $\zv_{+}$ under the
boundary condition $\bar x$ on $\zv_{-}=\zv\setminus\zv_+$.
Take, for example,
$\bar x = +$ for the Ising model with a positive magnetic
field. This yields the ``Basuev'' state on $\zv_{+}$.
\newline {\bf 4.}
In fact, meaningful and nontrivial results
requiring the full strength
of all the forthcoming constructions can be formulated
even without taking the thermodynamic limit,
for a fixed {\it finite\/} volume $\Lambda$
(imagine the cardinality $|\Lambda | = 10^{27}$ !), with suitable
boundary conditions.
However, in such a case it is of course natural to study also
a torus with periodic boundary conditions.
Though we do not work out here the (topological) modifications
needed to carry on our study also to the case of
periodic boundary conditions, it seems that only
minor parts of the text should be modified or
replaced by another arguments (for example, the
parts of the text where the lexicographic order
is used)
to apply our method.
\vskip1mm
\head 1. Stratified configurations, Hamiltonians and States
\endhead
For any $u\in\zw$ consider the shift \
$ U \equiv \{t \mapsto t+(u,0)\}\, : \zv\rightarrow\zv $
and correspondingly define the shifts
$ \{x \mapsto U(x)\}\, : \ S^{A}\rightarrow S^{U(A)}
$
where $U(x)=\tilde x$ has coordinates $\tilde x_{t+u}=x_t$,
$t\in A$ and
$
\Phi _{A} \mapsto (U\Phi) _{U(A)}
$ \
where $(U\Phi) _{U(A)}(U(x_{A}))
= \Phi _{A}(x_{A})$.
Say that a configuration $x$ is {\it stratified\/} (or
{\it horizontally\/} {\it invariant\/}) if
$
U(x)=x\quad \text{for each } u\in\zw $ where we
identify $\zw$ as a subset
$ \zw =
\{(t_1,\dots,t_{\nu -1},0)\} \subset \zv\,.
$
\definition{Notation} We denote by $\es\subset \ex$ the collection of
{\it all stratified configurations\/}.\enddefinition
Analogously we define the notion of a {\it stratified
Hamiltonian\/} $H=\{ \Phi _{A} \}$ and a
{\it stratified\/}\ (Gibbs) {\it measure\/} $\mu $
by requiring, for each $u \in \zw$,
$$
\{ (U\Phi)_{A}\}=\{ \Phi _{A}\}\,,\,\,\,U(\mu )=\mu.
$$
\remark{ Notes} {\bf 1}.
These configurations will be the ``local
ground states'' of our model.
Of course only {\it some\/} of them will
``deserve'' this name. Analogously, in the traditional
Pirogov -- Sinai situation, only some of the constant (resp\. periodic)
configurations ``deserve'' the name of ``local ground
state''. However, it is often problematic to separate these
``true local ground states'' from the other horizontally
invariant (analogously, translation invariant resp\.
periodic) configurations.
Moreover, the ``reference family'' $\es$ will have to be
chosen such that whenever $x \in \es$
locally (i.e. $x$ is equal to some $ y \in \es$
in the $r$ --neighborhood of each point)
then we have (``globally'') $x \in \es$.
Also, nonexistence
of a substantial energetic barrier between the
``true'' local ground states
and the remaining elements of $\es$ requires some care in
the formulation of
the Peierls condition, see below. The simplest
solution in such a situation
is to
choose the reference family $\es$ as big as above, and to
look for energetical barriers only between $\es$ and
configurations which are {\it not\/} stratified.
\footnote{This is an interesting methodological point even for the
ordinary Pirogov -- Sinai theory. Namely, we
now suggest to consider the family of
{\it all\/} translation invariant configurations
as the ``reference family''of configurations in the
ordinary Pirogov --Sinai setting. Such an approach leads, in fact, to a
sharper and clearer formulation of the Peierls condition. (See
below in (2.14) for our analogous situation.)}
\newline
{\bf 2.}
The framework when all the local ground states of the model are
are chosen among the stratified ones
(analogously, compare the choice
of translation invariant configurations
in the ordinary Pirogov -- Sinai theory)
seems at the first sight to be too narrow in the situations (like Ising
antiferromagnet) where {\it periodical\/} (local) ground
states occur. However, it is easily seen that periodical
resp\. horizontally periodical configurations can be
converted to constant resp\. stratified ones by taking suitable
{\it blockspin transformation\/} (over the periods)
and so the setting we
introduce here is sufficiently general.
\newline
{\bf 3.}
The fact that we are selecting {\it several\/} configurations
(in fact the whole
family $\es$)
as the ``reference'' ones
-- expecting that some of these configurations may
(possibly, under
suitable adjustment of the Hamiltonian) give rise to corresponding
Gibbs states -- suggests that our interest lies in the situations
where phase transitions {\it may occur\/}.
Thus, the possible ``degeneracy of the ground state''
\footnote{
By the {\it degeneracy\/} of a ground state one usually means
that {\it several\/} ground states exist
for a given Hamiltonian. Here, we are looking
for ground states (resp. more generally for
``local ground states'')
only among the elements of $\es$,
i.e. we are looking for configurations $y \in \es$ such that
$\sum_A (\Phi_A(x_A) -\Phi_A(y_A)) > 0 $ whenever $x$ differs from
$y$ on a set
whose vertical size is finite (resp. ``not too big'').}
is the situation of our interest.
Though in most situations we will have to deal with
only {\it one\/} Gibbs state corresponding to a given
Hamiltonian, we want to have a theory dealing at the same time
with the situations
of phase coexistence.
This requirement distinguishes the \ps\
theory from the methods focused on the study
of the {\it phase uniqueness\/} region. Namely,
recall that in the region
of phase uniqueness other and well developed methods of study
(based essentially on the Dobrushin' s
unicity theorem and on the later
investigation of the ``complete analyticity'' properties
by \cite{DSA}) are available.
On the other hand, in the regions
where phase coexistence is expected apparently no serious alternative
to the \ps\
theory exists reaching a comparable level of generality
and universality of its applications.
\endremark
\vskip1mm
\head 2. Precontours and Admissible Systems of Precontours
\endhead
\definition{Stratified points of a configuration $x$}
Given a configuration $x\in\ex$ say that a point
$t\in\zv$ is a {\it stratified\/} point of $x$, more precisely {\it
$y$-stratified\/} (for $y\in\es$) point of $x$, \ if
the equality $
x_{\tilde t}=y_{\tilde t}
$
holds for each $\tilde t\in\zv$ such that $||\tilde t-t||_{
\infty} \leq r$.
\enddefinition
\remark{Note}This is an analogy of the notion of a correct point of the
ordinary Pirogov -- Sinai theory; $r$ is the range of the
interactions in the given Hamiltonian (2.2).
\endremark
\definition{Diluted configurations,
external colour of a configuration}
Say that $\Lambda\subset \zv$ is a {\it standard set\/}
if all the sets
$C_n=\{t\in\zv;\, t_\nu=n\}\setminus\Lambda$
are {\it simply connected\/},
i.e. if any two points
$t,t' \in \Lambda ^{c}$, resp. $ \Lambda$, with the same last
coordinate $t_{\nu}=t_{\nu}'$ can be connected by a
``horizontal'' (keeping the last coordinate intact)
connected path in $\Lambda ^c $, resp. $ \Lambda$.
A configuration $x\in\ex$ will be called {\it
$y$ -- diluted\/}, for $y \in \es$, if there is some set \
$
\ee\subset\zv
$ \
whose all connected
components are finite standard sets
(an infinite number of components is allowed)
and such
that all the points of $\ee^c$ are $y$ -- stratified.
\footnote{This traditional
notion of \ps \ theory is {\it not\/} best suited for our purposes. It
will
be {\it strenghtened\/} below.
See the notion of an {\it isolation
(and strict diluteness)\/} below in (2.5) (and (2.18)).}
The value $y$ will be
called the {\it external colour\/} of $x$, denoted
by $y=x^{\text{ext}}$.
\enddefinition
\definition {Precontours} For any configuration $x$
denote by $B(x)$ the collection of
all its nonstratified (i.e\. stratified for {\it no\/}
$y\in\es$) points. If $C$ is a {\it finite\/} connected component of
$B(x)$ then the pair
$$
\gb =(C,x_C)
$$
will be called the {\it precontour of $x$\/} and we will write
$
C=\supp \gb
$.
We will say that $\gb$ is a {\it precontour\/} if it is a precontour
of {\it some\/} configuration $x \in \ex$. \enddefinition
\definition {Admissible systems of precontours}
By an {\it admissible system of precontours\/}
we will mean
any system $\Cal D =\{\Gamma _i\}$ of precontours which is
a collection of {\it all\/} precontours of {\it some\/}
configuration $x $. \footnote{In the next
50 pages, we will work
exclusively with {\it finite\/} contour systems
(in a finite volume $\Lambda$).}\enddefinition
\remark{Note}
The configuration $x$
is uniquely
determined by $\Cal D $ only in all the horizontal planes
intersecting $\Cal D $\,, otherwise it will be given
by the boundary condition
$y \in \es$
given
outside of
the volume in which we will be
working. It will be \footnote{ This is a slight abuse of notations but we will see later that
the geometrical notions defined below with the help
of $x_{\Cal D }$ will
not depend, in fact, on the ambiguity in
the choice of $x_{\Cal D }$.}
denoted by $x= x_{\Cal D,y}$ or (usually)
shortly by
$x = x_{\Cal D }$; the external colour $y$
of $x$ will be denoted also as $x^{\text{ext}}$
and the external colour of
$x_{\Cal D}$ as
$x^{\text{ext}}_{\Cal D}$.
Thus, we represent each diluted configuration $x$ or $x_{\Lambda}$
as
$x = (y, \Cal D)$ where $\Cal D$ is the system of
precontours of $x$ or $x_{\Lambda}$
and $y$ is an additional information about the boundary
condition on $\Lambda^c$ resp. on ``the value in infinity''
if we are in infinite volume. The value of $y$ will be
usually clear from the context and therefore
mostly omitted in the notations; we will often speak about
a configuration $x = \Cal D$, etc. \endremark
\vskip1mm
\head 3. Contours
\endhead
One could characterize the \ps \
theory as a play with the concept
of a ``contour''; the latter being understand as some barrier
between different ``correct'' (stratified in the context
of the presented paper) regimes. The very
idea of the ``removal'' of a contour $\gb$ -- together
with the comparison of the partition function of
the original regime ``inside'' $\gb$
with the partition function of the
new regime (which appears after the removal of $\gb$) plays a
decisive role there; and this must be supplemented by suitable
combinatorial
bound for the number (something like $C^N$) of different shapes of
contours of a given cardinality $N$.
However, it would be sometimes problematic to realize the idea
of a ``removal'' of $\gb$ for a precontour $\gb$ which has a topologically
complicated
shape. As an example of such complicated shapes one can
have in mind something like a ``bumerang'' or, say, a
``ring'' over such a bumerang or ``fingers'' of one
precontour penetrating the ``interiors''
of other precontours. Surely
the concept of an interior of these objects deserves
a more careful definition.
However, precontours are {\it not suitable objects
to do that\/}.
One solution of this problem is the concept of a wall
introduced in [D].
The idea of the
Dobrushin wall suggests some {\it glueing of precontours
together\/}
as a means to clarify the
concept of an interior of such a wall
(which is
defined as a suitable conglomerate of precontours).
However, when glueing these topologically difficult
objects together one has to keep in mind another highly
desired property of contours which should be fulfilled -- namely
their ``connectedness'' (whatever one could
have in mind by this).
The concept of a Dobrushin wall
was used also by several other authors later,
notably in \cite{HKZ} and \cite{HZ} (while in
\cite{MS} the geometrical notions used for the description
of the interface were more different).
Such an approach resolves the
problems mentioned above; however it seems to be applicable
(by considering
the projection of the situation appearing at the interface to
the sublattice $\zet^{\nu -1}$)
only in the special situation
of {\it one\/} interface.
This construction can be hardly transferred
to the situations where two or more parallel ceilings appear, and
therefore we will {\it not\/} follow such an approach.
Of course, our abandoning of the concept of a wall
is meant on technical level only, for
topologically ``obscure'', nontypical representatives
of the idea of a wall. Nevertheless the concept of a wall
as a connected conglomerate
of precontours defined with the
help of a $\nu -1$ dimensional projection will be lost, replaced by a more
general notion of a contour below.
This will cause some problems,
especially when the necessity check some
``connectedness'' of the contours
will finally emerge in
section III. Namely, we will have to control
the convergence of cluster expansion formulas developed
there; some combinatorial bounds for
the number of possible contours (more generally, for the
number of``clusters of contours'')
will be necessary.
Remember that precontours
{\it are\/} connected by their very definition, but unfortunately
there is {\it no\/} natural hierarchy between them.
We will see later in section III
that the problem of connectedness of our more general
(``less connected'' than usually) contours
can be managed with some
effort. See the section ``Tight sets'', part III.
There is no need to discuss this problem just {\it now\/}
-- and so we give the definition
of a contour below without explaining here the true
advantages
of such a choice.
This will become
clear later.
In fact, we could live without contours in the forthcoming
text, keeping only the notion of a compatible system of precontours
and having in mind that operations like the
``removal'' of $\gb$ should be never applied to precontours
or systems of
precontours which are not compatible.
However, the concept of an isolation introduced
below {\it will\/} be extremely important
for us and the notion of a contour is just its obvious
byproduct,
so why not to keep such a notion, traditional
in the \ps \ theory
\footnote{ In the previous versions of the paper we had a much
more restrictive notion of a contour $\gb$ -- being defined as some
conglomerate of precontours
whose any subconglomerate is ``split'' (in the topological
sense, in a suitable horizontally invariant section of $\gb$)
by the rest of $\gb$. The reader will see in part III that
our contours really are some ``connected
conglomerates of precontours'', though in a different
(and more general) sense
than in \cite{D}.
See the section ``Tight sets''.} also here.
\definition {Contours}
Say that a subset $S \subset T$ of $T \subset \zv$
is {\it isolated in\/} $T$ (or {\it from\/} $T\setminus S$) if
$$ \dist ( S,T \setminus S) ) \geq
\diam S.\tag 2.5
$$
By the {\it diameter of $S$\/} we will mean, everywhere
in the following, the $l_{\infty}$ diameter of
$S$. Say that $T$ is {\it tight\/} if it has no
isolated subset. An
admissible system $\gb =\{\gb_i\}$ of precontours will be called
a {\it contour\/} if
the set $ \cup_i \supp \gb_i$, denoted also as
$\supp \gb $, is tight.
Denote further by $\ext \Gammab$ the collection of
all points of $(\supp \Gammab)^c $ which can be accessed
from infinity by some vertically constant
(i.e. keeping the last coordinate
intact),
correct (in the sense that each point of the path
is a stratified one) connected path. Define also
the volume (having only a minor importance in our
approach!)
$ V(\Gammab)=(\ext\Gammab)^c \,.
$
We will say that $\gb$ is
a contour {\it in a volume\/} $\Lambda$ if
$\gb$ {\it is isolated from\/} $\Lambda^c$ i.e. if
$\supp \gb$ is isolated from $\Lambda^c$. We will
denote this fact by $\supp \gb \sqsubset \Lambda$
(we introduce such a special notation $ S \sqsubset
\Lambda$ only for sets $S$ which
are supports of contours)
or $ \gb \sqsubset \Lambda$. So we have
$$ \gb \sqsubset
\Lambda \ \ \ \Longleftrightarrow \ \ \
\dist(\Lambda^c,\supp \gb)\geq \diam \supp \gb
\ \ \ \ \Longleftrightarrow \ \ \ \vv \subset \Lambda
\tag 2.6 $$
where the important notion of ``protecting zone'' $\vv$ of
a contour $\gb$
is defined as
\footnote{ However, if $\Cal D$ is an admissible
system which is {\it not\/} a single contour then we
put (compare (2.16))
$\vvd = \cup_i \vvi $ where $\{\gb_i\}$ is the
representation (see Proposition
below) of $\Cal D$ in
terms of its contours.}
$$ \vv = \{ t \in \zv:\ \dist(t,\supp \gb)
\leq \diam \supp \gb\} .\tag 2.7$$
(We have
$V(\gb) \subset \vv$, as simple inspection
of any horizontal slice of $V(\gb)$ shows.)
\enddefinition
\definition{Admissible families of contours}
A collection of contours $\Cal D =\{\gb_i\}$ will be called admissible
if each $\supp \gb_i$ is isolated from
\ \ $ \bigcup_{j \ne i: \diam \gb_i \leq
\diam \gb_j } \supp \gb_j.$
Contours of the system
$\Cal D$ which are isolated from the remainder
will be called
{\it interior\/} contours of $\Cal D$.
\enddefinition
\proclaim{Proposition}
Any admissible system $\Cal D =\{ \gb_i\}$ of precontours
can be uniquely decomposed to an admissible
system $\tilde \Cal D =\{\tilde \gb_j\}$ of contours.
\endproclaim
This is immediate if we realize
what the
interior {\it contours\/} of $\Cal D$ are: they are defined
as admissible subcollections of $\Cal D$ containing no
smaller isolated
subcollection. Notice that an intersection of
any two isolated
subcollections of $\Cal D$ is isolated
resp. empty, if moreover
one of these subcollections is minimal. Thus two different
interior
contours
are always disjoint and mutually isolated.
Having such (uniquely defined) interior
contours we can remove them (replacing each such
interior contour $\tilde \gb$
by its
corresponding external colour) and one obtains a smaller
admissible system where the new, ``bigger'' interior contours
can be again uniquely determined etc. It is straightforward
\footnote{ Notice that if an interior contour
$\tilde \gb$ of $\Cal D$
with a smallest possible diameter
is removed from $\Cal D$, then no
interior contour
$\tilde \gb'$ with a smaller diameter than $ \tilde \gb$
can be found in the
remainder $\Cal D \setminus \gb$!} to establish the isolation
properties of such a system.
\remark{Notes}
0. We stress that
``admissibility'' for a system $\{\gb_i\}$
of precontours means
just a {\it nonconflicting\/} prescription
of ``colours outside of $\cup_i \supp \gb_i$ ''
while admissibility of a system of {\it contours\/}
requires also the
isolation of any $\tilde \gb_i$
from all contours
$\tilde
\gb_j:\tilde \gb_i \prec \tilde \gb_j$.
1. The volume $V(\gb)$ is the intersection of
all standard volumes containing $\supp\gb$.
In the rest of the paper (the exception will be the proof of our
Main Theorem; actually this will be really relevant only
when developing some explanatory notes
interpreting
the detailed structure of the phases constructed by Main Theorem)
we will work only with
{\it finite\/}, often standard,
volumes $\Lambda$. In fact, one could restrict oneself only to
the {\it cubes\/} $\Lambda$ in most following
considerations. The boundary
condition
$y \in \es$ will be always
given on the boundary of $\Lambda^c$ .
(Thus we will work
with diluted configurations having a finite
number of precontours only.)
2.
Let us comment once again the topological
problems we meet here, in comparison to the ordinary
translation invariant \ps \ theory.
In the ordinary translation invariant theory, the elements of
the ``reference set'' $\es$ are constant
configurations. In such a case one immediately realizes that precontours
(called there contours) are
``crusted'', in the sense that the (only!) infinite component
of $(\supp\Gamma )^{c}$ (which will be called the exterior of $\gb$)
satisfies the property that
all its points are ``$y$ -- correct'' where $y$ is the
external colour of $\Gamma $. Even more importantly,
the interior of $\gb$ (the union of the remaining components)
is always ``disconnected'' from the exterior of $\gb$,
for any precontour $\gb$.
This enables to construct ``telescopic equations''
relating the diluted partition function in a given volume $\Lambda$
to the ``crystallic'' (see e.g. [S])
partition functions of the external contours appearing in $\Lambda$,
and therefore to the
diluted partition functions of the {\it interiors\/} of these contours.
This is {\it not \/} so here, where
precontours can have the shape of Dobrushin's ``walls''
separating
various types of ``ceilings'' i.e. various horizontally
invariant configurations in our case. Even if the
notion of an exterior resp. interior of a contour is well defined
these two volumes
are usually not separated by an ``inpenetrable crust'' like
in the ordinary translation invariant situations.
\head 4. Expression of the Hamiltonian and the Peierls
Condition \endhead
For any stratified configuration $y\in\es$ \ we define now its
``density of energy'' at $t\in \zv $ :
$$
e_t(y)=\sum_{A\ni t}\Phi _A(y_A) |A|^{-1} \,.
\tag
2.8 $$
Given a (pre)contour $\gb$ one would like to define also a quantity
having the meaning of the ``(pre)contour energy''. One could think,
for {\it contours\/},
for example about the
``energy excess'' of $H(x_{\gb})$, where $x_{\gb}$ denotes
\footnote{We might say that the information about a
contour $\gb$ consists of 1)
information about
its support $\supp \gb$ and 2) information
about a configuration $x_{\gb}$. Here, we need to
know only the value of $x_{\gb}$ restricted
to ${\supp \gb}$.
However, quantities like $F(\gb)$
will require the knowledge of $x_{\gb}$ in the whole
$\vv$.}
the configuration having $\gb$ as its {\it only\/} contour,
with respect to
``something like $H(x_{\ext})$'' where $x_{\ext}$ denotes
the ``external colour of $\gb$''.
However, such a straightforward approach to the definition of an energy
of a contour is reasonable only in the
very special cases (like the Ising model
with zero magnetic field) when the density of energy inside $\gb$ is
the {\it same\/} as outside of $\gb$, at {\it any} horizontal level.
Otherwise it will be useful (to keep the interpretation
of a contour energy as a quantity which is ``localized on $\supp \gb$ '')
to replace the quantity
$H(x_{\gb})-H(x_{\ext})$ just mentioned
\footnote{This will be, of course, also {\it very\/}
important later --
see (2.19)!} by the following quantity
(perhaps too formally defined at first
sight):
Let us first extend the notation $e_t(x)$ for any (even {\it nonstratified\/}
in $t$)
$x$ by putting
$$
e_t(x)=e_t(\hat x)\,; \quad
t=(t_1,t_2,\dots,t_{\nu})
\tag 2.9 $$
where $\hat x$ is the {\it stratified continuation\/} of the
vertical section
$\{x_{(t_1,\dots,t_{\nu-1},(\cdot))} \}$.
Now define, for any finite $G \subset \zv$ (or, more
generally, such that any vertical section of $G$ has
finite components only)
and any diluted configuration $x$ such that all points of
$G^c$ are {\it stratified\/} ones,
the auxiliary configuration $x^{\text{best}}_G$
{\it minimizing\/} the sum in variable $z$ below,
where $y$ denotes the
``external colour'' of $x$ (notice that the terms of the
infinite sum below are equal to zero outside of $G$):
$$
\sum_{t\in\zv}(e_t(z) -e_t(y)) \tag 2.9'
$$
under the condition that the variable $z$ equals to
$x$ on the set $G^c$ and also on the
set \ $\partial_r G=\{t \in G,\
\dist(t,G^c)\leq r\}$.
Write $x_{\gb}^{\text{best}}$ instead
of $(x_{\gb})^{\text{best}}_{\supp \gb}$ where $x_{\gb}
=x_{\gb,y}$ ($y$ is a boundary condition)
is a configuration having $\gb$ as its single contour.
\footnote{ One should not try to interpret such an
``artificially chosen'' configuration
$x_G^{\text{best}}$ or
$x_{\gb}^{\text{best}}$
in the points where it is {\it not\/} stratified. In particular
the choice of $x=x_{\gb}^{\text{best}}$
should not be confused with the effort (natural perhaps, but
inconvenient technically) to
minimize $H(x)$ from (2.2) under the restrictions above!}
Put
$$
E (\gb)=H(x_{\gb})-\sum_{t\in\zv}
e_t(x_{\gb}^{\text{best}}) \ \ \ \text{where} \
\ \ H(x) = \sum_{A \subset \zv}
\Phi_{A}(x_A).
\tag 2.10
$$
Of course this is again only a formal expression. However,
the terms in the sums on the right hand side of the equation for $E(\gb)$
can be obviously reorganized such that
a sum with only a finite number of nonzero terms is
obtained. Namely one can rewrite (2.10) in the
following equivalent and more precise way, pouze footnote
(even for {\it pre\/}contours; but then (2.10) must be
written
more carefully, not using (generally nonexistent)
$ y = x_{\gb}^{\text{ext}}$)
$$ E(\gb) =\sum_{A \subset \supp \gb} \bigl(
\Phi_A(\gb_A) -
\sum_p \frac{|A\cap p|}{|A|} \
\Phi_A(( \hat x_{p}^{\text{best}})_A) \bigr) \tag 2.11 $$
where $\gb_A$ denotes the restriction
of $\gb$ to $A$, the second sum is over
all ``verticals'' $p =\{t_1,\dots,t_{\nu -1}, (\cdot)\}$
intersecting $\supp \gb$
and $\hat x_{p}^{\text{best}}
$ denotes the horizontally invariant
extension of the restriction of $x_{\gb}^{\text{best}}$
to
$p$. In the (very special!) case of
a {\it stratified\/} ``reference''configuration
$x^{\text{best}}_{\gb}$
the
formula (2.11) can be written
in the following, more simple and more
transparent way:
$$ E(\gb) =\sum_{A \subset \supp \gb} \bigl(
\Phi_A(\gb_A) -
\Phi_A(( \hat x_{\gb}^{\text{best}})_A) \bigr). \tag 2.11' $$
Then we have the following expression of the Hamiltonian
(2.2):
\proclaim{Theorem 1}
Let $x$ be a configuration such that all its
(pre)contours have supports in a given finite
\footnote{$\Lambda$
will always denote a finite set in the sequel.
The constant $C$ in the formula ( 2.12)
does not affect Gibbs measure defined by (2.12) in
a fixed volume $\Lambda$. We
will put $C =0$ in the following considerations.}set
$\Lambda$.
Then there is a constant $C =C(\Lambda,y), y =
x^{\text{ext}}$ such that
$$
H(x_\Lambda |x_{\Lambda^c})=\sum_{t\in\Lambda }
\hat e_t(x) +
\sum_{\gb \in \Cal D} E(\gb) +C=
\sum_{t\in\Lambda }
e_t(x^{\text{best}}_{\Cal D}) +
\sum_{\gb \in \Cal D} E(\gb) + C
\tag 2.12 $$
where $\Cal D$ denotes the system of all precontours of $x$
and where we are using also the notation
$ \hat e_t(x)$
for $
e_t(x^{\text{best}}_{\Cal D})$; (see (2.9) and (2.9'))
.
\endproclaim
One has to prove that
(2.2)
with
the notations (2.9), (2.11)
imply (2.12).
Instead
of giving here the details of the (noncomplicated!) arithmetics
of this argument
we notice only
that (2.12)
follows immediately from the formal expression
(2.10). Notice also
from (2.11)
that $E(\gb)$ is a local quantity, depending on $\gb$ only
and also an additive one:
$$E(\gb\cup\gb')=E(\gb)+E(\gb') . \tag 2.13 $$
\definition {Peierls condition}
In the following we will assume that for any finite $G \subset
\zv$ and for
any stratified configuration $y\in\es$ the
following inequality
holds with a large
\footnote{The requirement on the
largeness of $\tau$ depends on the dimension $\nu$.
For translation invariant problems and $\nu =2 $, one could possibly reach something like
$\tau > 5$. Unfortunately, our estimates will not be
apparently always
the strongest possible and our actual requirement on $\tau$
will be much stronger,
something like, say, $\tau > 50$.}
$\tau>0$ :
$$
\sum
\Sb \gb:\ \supp \gb=G \ \& \ x_{\gb}^{\ext}=y\endSb
\exp(-E(\gb)) \leq \exp(-\tau|G|)
\tag 2.14
$$
where the summation is over all {\it contours\/} having the support $G$.
(We recall that we include the inverse temperature into the
Hamiltonian and therefore $\tau$ is of the order of the
inverse temperature in most examples.)\enddefinition
\remark{Note}
In practice, one can establish (2.14)
with the help of inequalities, for
some $\tau^* \geq \tau$,
$$ E(\gb) \geq \tau^* |G|. \tag
2.15 $$
The belief is that (like in the usual
translation invariant situations) such a condition
{\it should hold in most usual situations\/};
in fact the counterexamples
of (translation invariant) models with
nonvalid Peierls condition
constructed by Pecherski \cite{PECH} and later by
Miekisz \cite{MIE} are rather nontrivial.
Concerning an effective constructive
criterion how to check the validity of
(2.14) or (2.15), one could suggest a suitable
modification of the
Holsztynski -- Slawny criterion
(see \cite{HS}, \cite{EFS}).
This question surely deserves further
study; specifically for stratified
ground states. We did not investigated this in detail yet.
\vskip1mm
\head 5. Strictly Diluted Partition Functions.
\endhead
\vskip1mm
Recall that in
addition to the volume $V(\gb)$ we have another (technically
much more important later) notion of $\vv$ which will be
needed for any contour $\gb$ (and also for some special,
``recolorable''
{\it systems\/} of contours --see part III, sections 4 - 5.)
The reason for replacing of $V(\gb)$ by
$\vv$ in definitions like
(3.19) will be understood only later.
\definition{Strictly diluted partition functions
$Z_{\updownarrow}^y(\Lambda)$}
Recall that for a contour $\gb$, $\vv$ is the union of points from
$\zv$ whose distance to $\supp \gb$ is not greater than $\diam \gb$.
If $\Cal D = \{\gb_i\}$ is an admissible
system of contours then
we define the volume $\vvd$ as
\footnote{ One could take a union
of ``protecting zones'' $V'(\gb)$
of {\it pre\/}contours of
$\Cal D$ here and everywhere. Once we decided
to introduce the concept of a contour,
such a modification would bring no advantage.
Also, one could argue that the
protecting zone $V'(\gb)$
is chosen unnecessarily big.
However this will make no harm, too.
In later sections of part III (namely in the construction of a ``skeleton''
of $\gb$) we will be a little bit more cautious
when tackling ambiguities analogous to those appearing in
(3.19). In fact, a neighborhood $V'$ of $V(\gb)$
up to a distance $\approx \log \diam V(\gb)$
would be sufficient here, too, but we do not care.}
$$\vvd = \cup_i \vvi. \tag 2.16$$
We denote, from now on, by
$ Z^y_{\updownarrow}(\Lambda)$
the ``strictly diluted'' partition function
\footnote{{\it This\/} partition function
will be our main
object and we will study it systematically
in the forthcoming text. In fact,
other interesting
partition functions $Z(\Lambda)$ (even those which can {\it not\/} be expressed
solely in the language of contours) can be easily
related to suitable ``strictly diluted'' partition
functions $Z^y_{\updownarrow}(\Lambda')$ of subvolumes
$\Lambda' \subset \Lambda$. It is also possible to
modify the forthcoming constructions to the case
of more general ``$y$ -- like'' boundary conditions;
in the final section of the paper
we will discuss modifications needed for the
study of such more general
partition functions.
However, especiallly for the case of stable
boundary conditions, this is
a minor point of the whole story, and the difference between
different partition functions mentioned above
is unessential. On the other hand, strictly diluted
partition functions are, of course,
{\it not\/} suited to {\it directly\/}
describe, without some
adaptations, also the {\it nonstable behaviour\/}
in a volume $\Lambda$
because very large (of the size
comparable to the size of $\Lambda$)
contours are {\it forbidden\/} here just by
definition. This problem will not bother us in
the present paper concentrated
on the stability properties of various $y \in \es$.}
$$ Z^y_{\updownarrow}(\Lambda )=\sum_{x_{\Lambda}}
\exp(-H(x_\Lambda |y_{\Lambda ^c}))
\tag 2.17 $$
where the sum is over all $x_{\Lambda}$ such that
all contours $\gb$ of the configuration (on $\zv$)
$x_{\Lambda} \cup y_{\Lambda^c}$
satisfy
$ \dist(\supp \gb,\Lambda ^c)\geq \diam \supp \gb $;
this condition will be
written also as \footnote{This is just a suitable
notation, like $A \cap \cap B = \emptyset$ below.
In general,
$A \sqcap B \ne B \sqcap A$ !}
$$ \supp \gb \ \sqcap \ \Lambda^c = \emptyset
\ \ \text{or} \ \ \supp \gb \sqsubset \Lambda. \tag 2.18 $$
In other words, we require that each
contour $\gb$ of $x_{\Lambda}$ would be isolated
from $\Lambda^c$. Let us introduce also another
notation $\Cal D \ssubset \Lambda$,
equivalently $\Cal D \cap \cap \Lambda^c = \emptyset$
used for {\it systems\/} $\Cal D$
of contours $\{\gb_i\}$ and defined by
$$ \supp \Cal D \ssubset \Lambda \ \ \text{iff}
\ \ \ \gb_i \sqsubset \Lambda . \tag 2.19$$
\enddefinition
\remark{Note}
For a comparison, introduce also the analogy
of usual, ``diluted''
partition
function
$$ Z^y(\Lambda )=\sum_{x_{\Lambda}}
\exp(-H(x_\Lambda |y_{\Lambda ^c}))
$$
where the summation is over all (``diluted'')
configurations,
whose contours satisfy the requirement, say
\footnote{ The precise requirement on the {\it
boundary\/} condition
is less important here; the real difference from usual
``diluted'' partition functions being in the changed
meaning of our very notion of a contour.}
$ \dist(V(\gb),\Lambda ^c)\geq 2 $.
Such a traditional formulation of ``diluteness'' would be less convenient
with respect to our approach
and it will {\it not\/} be used later. \endremark
The rest of this paper (and the essence of the Pirogov -- Sinai theory in its
presented version) consists of our effort to {\it expand\/}
the considered diluted partition functions (2.17)
as far as it is possible or reasonable;
then we
deduce some important corollaries from them,
like the estimate of
the probability to find the
external configuration $y$
in a given point of $\zv$, in the Gibbs state
determined by
the boundary condition $y \in \es$. Namely,
the complementary event
will be shwn to have a probability of the order
$\exp(-C\tau)$.
The attempt to expand partition function (2.17) can be based
also on
the older idea of a contour model \cite{PS} (or of a metastable
contour model \cite{Z}). Though that notion in fact
almost disappeared from the presented version of Pirogov --
Sinai theory (instead
of speaking about ``metastable contour models''
we will work, in fact, only with {\it expansions\/} of
the partition functions
of the ``metastable submodels of the given model'')
it is useful to start with some intuitive arguments
suggesting the introduction of the
basic notion of a {\it contour functional\/}. This is just the
introduction to the later
technical constructions.
We will see that the very notion of a contour functional (\cite{S})
survived in our approach
(in contrary to
the idea of a contour model) and it is still, of course,
of a central importance!
\head 6. Abstract \ps
\ Models and Their Gibbs States. \endhead \vskip1mm
\definition{Strictly diluted configurations}
By this we mean a configuration
$x = (x^{\text{best}},\Cal D) $
where $\Cal D$ is an admissible
contour system.
If we are in a volume $\Lambda$ we require also that
$\vv \subset \Lambda$ would hold
for all the contours $\gb$ of $\Cal D$.
More precisely, by a configuration we mean
a pair $(x^{\text{best}}, \Cal D)$ where
the first term $x^{\text{best}} = x^{\text{best}}_{\supp
\Cal D}$ (recall that $x$ is stratified in all points of $(\supp \Cal D)^c$)
contains both the information about
the local ``colour''(the particular value from $\es$)
of the boundary of contours as well
as about the boundary condition on $\Lambda^c$;
and $\Cal D$ is a ``matching compatible'' system
of contours.
Contours $\gb$ and their energies $E(\gb)$
are either defined as above from some
``physical model'' (2.2) or are given
\footnote{
Such a preparation of the model (i.e. its
conversion to the form (2.12) by a suitable definition
of the notion of a (pre)contour) is not at all unique and
in some cases, there is even no ``most natural way'' how to do it, for
a given concrete model. One
can adapt the general scheme given above
in various ways for various concrete
situations -- by modifying the concept of a
``stratified point'', for example, in broader terms even by defining
contours as objects quite different from those we constructed here.
Remember, for example, that in the theory of translation
invariant
low temperature Ising models, contours are traditionally defined
as some selfavoiding {\it paths} (or surfaces in dimension
$3$) defined by plaquettes from the
the dual lattice. We leave to the reader to specify all the
details
of the analogy of an expression (2.12) in such a case
(then $x^{\text{best}}$ is just a ``best possible''
{\it constant configuration\/}
if translation invariant situation is considered)
and also in the stratified case.
However, these details are rather
irrelevant for the general strategy of the theory.}
as some abstract entities satisfying (2.14). In the latter
case, only the {\it last relation\/} of (2.12)
has sense, of course. Analogously, we can treat
the densities $e_t(y)$ as some abstract local
functions of $y \in \es$, having {\it no\/}
apriori relation to the energies $E(\gb)$.
Such an abstract approach will turn out to be very
useful later and in the following, we will work
{\it exclusively\/}
\footnote{Except of some explanatory remarks interpreting
the meaning of our Main Theorem, section III, 9.} with
configurations
allowing the
expression (2.12) and satisfying (2.14)
with a large $\tau$.
It is generally advisable
to develop the theory in
such an abstract setting (2.12),
with the Peierls condition (2.14) being established.
The reformulation of the original model to the language
(2.12) can be considered as a suitable
``preparation'' of the given ``physical'' model
(in some situations, verification of
the Peierls condition may be a nontrivial task!);
the Pirogov -- Sinai
theory actually {\it starts with the setting \/}
(2.12) \ \& \ (2.14). \enddefinition
\definition{ Gibbs states of abstract \ps \ models}
In finite volumes, by an $y$--th Gibbs state
on the space $\ex(\Lambda,y), y \in \es$
(of all $y$ -- strictly diluted configurations
in $\Lambda$) we will mean the probability measure,
denoted by $P_{\Lambda}^y $, determined by
the hamiltonian (2.12)
(recall that we can take $C=0$ there)
with the partition function
(2.17).
\enddefinition
What are infinite volume limits of such measures?
(See also \cite{S} for an additional information -- for
a case of contour models). Shortly speaking, take the following
class
of configurations defined on the whole volume
$\zv$. Consider all the pairs
$(x^{\text{best}}, \Cal D)$
where $x^{\text{best}}$ is defined as before
and $\Cal D$
satisfies (e.g.) the following condition.
\definition{Locally finite contour systems $\Cal D$ on $\zv$}
Say that an admissible contour system $\Cal D$ is
locally finite, resp. $I$ -- finite
if for each $t \in \zv$ the collection of
contours $\gb \in \Cal D$ such that $\gb$ is ``sufficiently close to
$t$ '', in the sense that \footnote{ Of course there is some
arbitrariness in such a requirement. ``Sufficiently close
to $t$''
is our substitute
for a more suspect (in the stratified case!)
concept of a ``contour $\gb$ containing $t$
in its interior''.}
$\gb \sqcap \{t\} \ne \emptyset$, is {\it finite\/},
resp. has a diameter at most $I(t)$.
Take $I(t) = |t|$ (for example).
Denote by $\ex^{\text{abstract}}_{
\text{fin}}$ the collection of all locally
finite abstract \ps \ configurations.
Denote by $X^{\text{abstract}}_{\Cal I} =
\cup_y X^{\text{abstract}}_{\Cal I,y}$ the collection of
configurations
which are $\tilde I$ -- finite for some shift
$\tilde I(t) = I(t+s)$ resp.
also have an external colour $y$. Each set
$X^{\text{abstract}}_{\Cal I,y}$ is thus the countable
union of compact sets $X^{\text{abstract}}_{I,y}$
of $I$ -- finite $y$ -- like configurations. \enddefinition
\definition{ Gibbs probabilities on
$\ex^{\text{abstract}}_{\text{fin}}$}
Any configuration space $\ex^{\text{abstract}}_{\Lambda,y}$
of all strictly diluted, with respect to the
boundary condition $y \in \es$, configurations
of the abstract \ps \ model
in a volume $\Lambda$ can be naturally embedded
into the space of all locally finite
$y$ -- like configurations $\ex^{\text{abstract}}_{\text{fin}, y}$.
In such a way, any probability measure
$P^y_{\Lambda}$ can be viewed
as a probability on $\ex^{\text{abstract}}_{\text{fin}, y}$.
Introduce the $\sigma$ -- algebra of subsets of
$\ex^{\text{abstract}}_{\text{fin}, y}$,
generated by all ``cylindrical'' events
$\{(x^{\text{best}},\Cal D): \gb \in \Cal D\}$ where $\gb$
is a contour system.
\footnote{ Surely, this is not an exhaustive list of
all cylidrical events on $\ex^{\text{abstract}}_{\text{fin}, y}$,
if we consider this configurations space as a subset of
the
original configuration space $\ex$. However, for the
Gibbs state on $\ex^{\text{abstract}}_{\text{fin}, y}$
one will see that the $\sigma$ -- algebra
generated by ``cylindrical'' sets above, completed
by sets of measure zero contains
also all the usual cylindrical sets,
like the specifications of
spins in finite sets $ \Lambda \subset
\zv$.}
\enddefinition
Consider now the
thermodynamic limits of these measures.
\definition{Thermodynamic limits and limit Gibbs states}
By a thermodynamic
limit over volumes $\Lambda \to
\zv$ we mean
just the very general requirement that
we have a {\it suitable sequence of volumes\/} $\Lambda_n$
such that the relation
the $\dist(t,\Lambda_n^c) \to \infty$
holds for some (equivalently, for each) $t \in \zv$.
Any thermodynamic limit (taken
in the above sense)
of probabilities $P^y_{\Lambda},
\ y \in \es$, understood in the sense of {\it convergence on
each cylindrical set\/} $\{(x^{\text{best}},\Cal D):\gb \in \Cal D\}$
will be called a {\it Gibbs
probability\/}
on $\ex^{\text{abstract}}_{\text{fin}}$,
respectively on $\ex^{\text{abstract}}_{\text{fin},y}
\subset \ex^{\text{abstract}}_{\text{fin}}$
if it exists
as a $\sigma$ -- additive probability on the space
$\ex^{\text{abstract}}_{\text{fin}}$,
resp. even on $\ex^{\text{abstract}}_{\text{fin},y}$.
The value $y \in \es$ will be called {\it stable\/}
if the limit has support in
$\ex^{\text{abstract}}_{\text{fin},y} \subset
\ex^{\text{abstract}}_{\text{fin}}$
(and not in some other
$\ex^{\text{abstract}}_{\text{fin},\tilde y},
\tilde y \in \es$).
See section 9,
part III for detailed information. \enddefinition
\remark{Notes} 1.
Some care concerning the
``allowed shape of $\Lambda$''
will be necessary -- to avoid possible
entropic repulsion/attraction effects which could
affect the picture. The use of so called
``conoidal`` sets (see the last section of
the paper,
boundaries of conoidal sets do not contain
flat pieces) makes the above effects negligible in
questions
concerning the ``stability of a stable $y$ also inside
of the volume $\Lambda$''.
This is important when
interpreting, in finite volumes,
the contents of our Main Theorem.
2. We will not discuss here the
details of the relation between the notion
of a Gibbs measure on the usual configuration space
$\ex$ (with Hamiltonian (2.2)) and Gibbs probabilities
on $\ex^{\text{abstract}}_{\text{fin}}$. Some explanatory
remarks will be given at the end
of section III but the problem of the
mutual relation between the usual notion of a
Gibbs measure on $\ex$
and Gibbs measures on suitable spaces
$\ex^{\text{abstract}}_{\text{fin},y}$ deserves
an additional study (perhaps in a suitably more general
setting covering also other applications of
\ps \ theory).\endremark
\vskip1mm
\head 7. The idea of a contour functional
\endhead
The basic task of the Pirogov -- Sinai theory is to determine those
configurations $y$ among the ``reference'' ones (reference
means stratified in our case)
which are stable in the sense that they give rise to a Gibbs
state, whose almost all configurations are some ``local''
perturbations of the considered reference (stratified)
configuration. (More precisely, they are ``$y$-- diluted''
almost surely.)
A useful and intuitively appealing tool to determine whether
a given configuration is ``stable'' is the construction of a
``metastable model'' \cite{Z}
around the given reference configuration.
To define such a metastable model one introduces
(\cite{S}, \cite{Z}) an auxiliary quantity called the
``contour functional'' $F(\gb)$
which can be interpreted as the ``work needed to install the given contour''
and
which is used for the
{\it test\/} whether $\gb$ is allowed as an
{\it external\/} contour of the metastable model.
To get an idea of the value of such a testing quantity let us start with
its simplified version, for the case when
``no contours within contours are allowed'':
Put
$$
F_0(\gb)=H(x_\gb)-H(x_\gb^{\ext})=E(\gb)-A_0(\gb)
\tag 2.20
$$
where
$$
A_0(\gb) = \sum_{t\in V(\gb)}
(e_t(x_{\gb}^{\ext})-e_t(x_{\gb}^{\text{best}})).
$$
This quantity is of course just a first approximation to the more
relevant quantity given at this moment only formally
by
$$
F_{\text{formal}}(\gb)=
\log Z^y_{\text{ref}}(\zv)-\log Z^{\gb}(\zv)
\tag 2.21
$$
where $y$ is the external colour of $\gb \ \ (y=x_{\gb}^{\ext})$
and $Z^{\gb}$ denotes the partition function ``over
all configurations on $\zv$ containing the contour
$\gb$''. The ``reference'' partition function
$Z^y_{\text{ref}}(\zv) $ is over all configurations
having the ``colour'' $y$ on $\supp \gb$ (i.e. being $y$ stratified there) and
satisfying
moreover the property that their contours do not ``touch''
$\supp \gb$.
Below we will define, by the
relations (3.21) and (3.22), a {\it rigorous counterpart\/} of this
quantity, which will play a very important role in the sequel.
For contours which are ``not very big'' the quantity
$F_0$ is a good approximation to $F_{\text{formal}}$.
It enlightens somehow the concept of a small (or recolorable)
contour
used below; the term $A_0(\gb)$ typically satisfies
an estimate like
$$
A_0(\gb)\leq C|V(\gb)|
\tag 2.22 $$
with $C$ much smaller than $\tau$ (imagine
the Ising model with a small external field; then $C$ has the
order of its intensity) and therefore if, say,
$
C|V(\gb)|\leq \frac{\tau}{2}|\supp \gb|
$
(this will surely hold for contours which are ``not too big'')
we have, from (2.15), the inequality
$$
F_0(\gb)\geq E(\gb)-C|V(\gb)|\geq
\frac{\hat \tau}{2} |\supp \gb|\,.
\tag 2.23 $$
We see that $y=x_{\gb}^{\ext}$ is really a ``local ground
state'' because installing of a ``not too big'' contour
{\it increases\/} its energy.
Unfortunately, it is not at all trivial to define
quantities like $F_{\text{formal}}$ rigorously in our
situation. While
analogous task is solved rather straightforwardly in other
applications of the Pirogov -- Sinai theory (where contours
are ``crusted''
in the sense that there is no dependence between events
inside and outside $\gb$), here the presence of ``penetrable ceilings''
(flat horizontal parts of boundaries of $V(\gb)$ which do
{\it not\/} belong to $\supp \gb$) causes
problems! These problems however can be solved with the help
of suitable {\it expansions\/} of the model,
and this is the main subject of
the forthcoming part of the paper.
\vskip1mm
\head 8. A summary of the main result
\endhead
Let us summarize once more our basic constructions applied so far
and outline our basic result.
Consider a model
(2.2). For any standard finite
volume $\Lambda$ , rewrite the Hamiltonian $H$
according to the formula (2.12), using the
``single spin interactions'' $e_t(y),y \in \es$ (more
generally $\hat e_t(x) =e_t(x^{\text{best}}), x \in \ex$) and the ``contour energies''
$E(\gb)$.
\footnote{Recall that while $e_t$ are the ``densities of energy of
$x$ at $t$'', the more general auxiliary
quantities $\hat e_t(x)$ do not have such a nice interpretation.
However, they enable us to formulate the Peierls condition
in a useful and
general way (2.14).}
\definition{An Outline of Main Theorem}
Assume that a Hamiltonian (2.2) is given satisfying the
Peierls condition (2.14) when reformulated by (2.12).
Then one can construct quantities
$$h_t(y) = e_t(y) - s_t(y) $$
such that in any volume $\Lambda$, for any boundary
condition $y \in \es$ given outside of $\Lambda$,
the probability $P^y_{\Lambda} [\gb]$ that a contour system
$\gb$ appears in $\Lambda$ is given by the formula
\footnote{We will return to (2.24) by formulating its precise
analogoue (3.93) at the end of the paper.}
$$ P^y[\gb] = \exp (-\tilde F(\gb)) \ \ \ ,
\ \ \ \tilde F(\gb) = E(\gb) - \sum_{t \in \Lambda}
(h_t(y) -h_t(z)) + \Delta \tag 2.24 $$
where $z$ denotes the local ``colour'' (value from $\es$)
induced by $\gb$ outside of $\supp \gb$ and where
the correction
term
$ \Delta =
\Delta (\gb,
\Lambda,y))
$ is usually {\it small\/}. More precisely
the bounds
$\Delta \leq \varepsilon |\supp
\gb|$ respectively
$|\Delta| \leq \varepsilon |\supp \gb|$
hold in the case
when $\gb$ is
strictly diluted
in $\Lambda$ and $y$ is ``not too unstable''
(see section 8, part III for a
more precise description) respectively (the stronger bound)
if moreover
no ``residual''
matching compatible
contours could be placed outside of $\gb$.
The quantity $h_t(y)$ may be interpreted as the ``density of the
free energy, at $t$,
of the $y$ -- th metastable state'' and it satisfies
the following property:
Whenever a choice
of $y$ is made such that $h_t(y)$ has its {\it ground value\/}
(in the sense that the sum $\sum_{t \in \zet}h_t(y)$ cannot
be lowered
if $y$ is changed on a strip of a finite width)
then the {\it existence of a Gibbs state\/} $P^y$
{\it of the ``$y$ --th type'' is guaranteed\/}. This will be called
the {\it stability\/} of $y$.
By the ``$y$ --th type'' we mean that $P^y$
almost any configuration looks, roughly speaking,
like
an ``infinite sea of
$y$ with small islands of perturbations''.
The terms $s_t(y)$ disappear in the zero temperature limit
$E(\gb) = \infty$.
\footnote{ Recall once again that we include the
inverse temperature into $H$ and therefore all our quantities
$E(\gb)$ (and also $e_t(y)$!)
are actually proportional to
the inverse temperature $\beta = T^{-1}$.
However, the
{\it difference\/}, for $t$ fixed, between various relevant
$e_t(y),y \in \es $
is typically of much smaller
order than $e_t(y)$.}
They
are given by suitable cluster expansion formulas, which
can be written as follows, according to the number
$k$ of contours forming a cluster
$\{\gb_i ,i =1,\dots,k\}$:
$$s_t(y) = \sum_k s^k_t(y) \ \ \
\text{where}\ \ \ s_t^1(y) =-\sum_{\gb: t \in \supp \gb}
|\supp \gb|^{-1} \exp (-F(\gb)) \tag 2.25
$$
the summation being over all ``recolorable''
contours $\gb$. The property of recolorability
will be defined in part III; all contours
will be automatically recolorable up to a certain size;
this size will depend
on the external ``colour'' $y$ of $\gb $ and will
become {\it infinite\/}
for $y$ stable.
The contour functional $F(\gb)$ mentioned in the
previous section has a
value
{\it very close\/} to the (nonlocal)
value $\tilde F(\gb)$
in (2.24); see however
(3.22) for an exact formula
showing also the {\it locality\/} of this notion
and see again (2.21) for its intuitive meaning.
The
remaining terms $s_t^2(y),\dots$ can be expressed analogously as
sums
$$ s_t^k(y) = \sum_{\{\gb_i\}: \cup_i \supp \gb_i \owns t} n_{\{\gb_i\}}
\prod_{i=1}^k \exp(-F(\gb_i)) \tag 2.25'$$
over suitable ``clusters''
\ $\{\gb_i\ ;\ i =1,\dots,k\}$ \
of recolorable contours mentioned in (2.25).
The
combinatorial coefficients $n_{\{\gb_i\}}$
do not depend, once $\gb_i$ are recolorable,
on the Hamiltonian $H$. They
are described below in the proof of Main
Theorem \footnote{The reader will realize later that
the terms forming the sum $s_t^k(y)$ are rather complicated but
really {\it very\/} quickly decaying -- the leading term
in these sums being $\exp(-C\tau N)$ where $N$ is the
sum of cardinalities of
the supports of contours
of a smallest possible cluster appearing in the sum
(2.26).}.
One could modify the expressions (2.25), (2.25') by allowing
sums over {\it all\/} $\gb$
(including the {\it non\/}recolorable ones)
with suitably defined, {\it large and fixed\/},
value of $F(\gb)$; see \cite{Z} for the concept
of the ``truncated
functional'' $\hat F(\gb)$. Then all the terms $s_t^k(y)$
would change continuously (with some additional
care they would change even differentiably or even more
smoothly; see \cite{ZA})
and moreover it can be proven that they
satisfy (if $F(\gb)/ |\supp \gb|$ is large!) the Lipschitz
condition, with a small constant, with respect to the
parameters $e_t(y)$ and $E(\gb)$ of the Hamiltonian (2.12).
Finally the partition function $Z^y_{\updownarrow \ \text{meta}}(\Lambda)$
corresponding to all diluted ``metastable'' configurations
in $\Lambda$
(see part III, section 9; the
subscript ``meta'' means {\it all\/} configurations
if $y$ is stable, otherwise some configurations -- namely
those
with
``external nonrecolorable contours''-- are excluded)
can be expressed via cluster expansion terms mentioned above,
for any standard volume $\Lambda$ as follows :
$$ \log Z^y_{\updownarrow \ \text{meta}}(\Lambda) =
-\sum_{t \in \Lambda} e_t(y) + \sum_{\{\gb_1,\dots,\gb_k\}}
n_{\{\gb_i\}} \prod_{i =1,\dots,k} \exp(-F(\gb_i)) =
\tag 2.26 $$ $$ =
-\sum_{t \in \Lambda} h_t(y) + \Delta(\Lambda)
\ \ \ \text{where} \ \ |\Delta(\Lambda)|
\leq \varepsilon |\partial \Lambda| \tag 2.26' $$
where the sum is over all clusters of recolorable
contours $\gb_i \sqsubset \Lambda$ and
$\varepsilon$ is small.
\footnote{Again (like in (2.24)), in comparison to the translation invariant
situations
the last estimate in (2.26')
is not very strong and will {\it not\/} be used below.
Namely
in general we may have $|\partial \Lambda| \gg
|\supp \gb| $ for a Dobrushin wall $\gb$ such
that $V(\gb) = \Lambda$, and then (2.26') is useless.
The expression by
cluster expansion in the first part of (2.26) will be
much stronger and more useful
in such situations.}
\enddefinition
\remark{Notes}
1. Apparently, there are {\it no other\/} stratified Gibbs states
than those corresponding to the ``minimal values of
$\sum_{t_{\nu}} h_{t_{\nu}}$''; however this is not proven
in this paper.
2. The functions $h_t(y)$ can be made in fact infinitely differentiable
if defined
more carefully; even local analyticity can be
achieved, like in \cite{ZA},
on the manifolds of parameters where the
collection of configurations
$y$ ``minimizing the value $\sum_{t_{\nu}}
h_{t_{\nu}}(y)$'' is fixed.
We plan a separate
paper devoted to both these questions.
\endremark
\vskip1mm
\head 9. Application: Stabilization of one dimensional
model in three dimensions
\endhead
It will be useful to formulate the following
corollary of our Main theorem which is better suited to concrete applications
(see the paper in preparation \cite{EMZ}).
Reasonable formulation of such a result can be
given only in the class of {\it infinite range\/} Hamiltonians
and we refer to part III for more details concerning the
(rather straightforward!) generalization of our Main Theorem
to the models with exponentially decaying interactions.
\definition{Essentially one dimensional interactions}
A horizontally
invariant, infinite range, indexed
by squares from $\zv$
interaction $\{\Phi_{\square}(x_{\square})\}$
which
is nonzero only if $x_{\square}$ is a restriction of some
{\it stratified\/} configuration will be called
essentially one dimensional. It can be identified
with an
interaction acting on $S^{\zet}$ and
indexed by intervals from $\zet$.
One should distinguish {\it two variants\/} of such a
notion
formulated 1) in the language of the abstract \ps \ model
resp. 2) in the language of the model (2.2).
Below
we use the (suitably adapted)
{\it first\/} approach which is
much more straightforward once
we have based our exposition on the expression (2.12).
Namely will moreover assume that the
interactions $\Phi_{\square}$ act only {\it outside of
all contours\/}, and that
they even may have a suitably modified value
(compared to the usual one, which is horizontally translation
invariant) in a distance, say, less than $\diam \square$
from some contour of the configuration.
Of course, for applications to real {\it spin\/} model
(2.2) one needs the approach 2) above --
allowing $\Phi_{\square}$
to act,
in a horizontally translation invariant way, also
{\it inside\/} of the contours -- if the restriction to $\square$
of the given configuration of contours
is stratified.
This is commented in more detail
in section 8, after the proof of Main Theorem.
A more detailed exposition of the implication
\ \ Main Theorem $\Rightarrow$ Corollary \
is needed (it is easier in the simplified
interpretation of $\Phi_{\square}$ we are using here)
see the end of Section III and
the paper \cite{EMZ}.
\enddefinition
\proclaim{Corollary}
Consider a model (2.12) with a hamiltonian, denoted
by $H_0$, satisfying
the Peierls condition (2.14).
Consider further a whole class of models on the same configuration space
$S^{\zv}$
with a Hamiltonian
$$ H = H_0 + H_1 \tag 2.27 $$ where $ H_1 =
H^{\varepsilon,\omega}_1 $ is from the
family $\Cal H^{\varepsilon,\omega} $ of all essentially
one dimensional
Hamiltonians defined by
interactions
$\{\Phi_{\square} \}$
acting outside of the contours of $H_0$ and
decaying like
$$ |\Phi_{\square} (x_{\square})|
\leq \ \varepsilon \ \omega^{\ \diam \square} , \tag 2.28$$
where $\omega, \varepsilon$ are chosen such that
$ \omega < 1$ and ${\varepsilon}/{1 -\omega}
\ll 1 $.
Consider the (uniquely defined!) decomposition of
the Hamiltonian $H_0$ into its
essentially one dimensional
\footnote{ If we have in mind stratified $y$ only.}
``ground part'' and the
``Peierls
part'':
$$ H_0 = H_G + H_P $$
where \ \ $e^{H_P}_t(y) \equiv 0$ for all $y$ and $E^{H_G}(\gb) \equiv 0$
for all $\gb$.
Then there is a mapping
\footnote{ We are not interested here at all in the
Peierls part of the image. Namely, when finding ground states
among various $y \in \es$,
the knowledge of the Peierls part (satisfying (2.14)) of the Hamiltonian
is irrelevant.}
$$ \{ H \equiv H_G +H_P + H_1
\mapsto H_G + H_1 +\tilde H_1 \}:H_0 +
\Cal H^{\varepsilon,\omega} \mapsto H_G + \Cal H^{\tilde
\varepsilon,\tilde \omega}
\tag 2.29
$$
where $\tilde \omega = \max(\omega,C\exp(-\tau))$
with suitable $C = C(\nu)$ \footnote{That is, $\tilde \omega = \omega$ if
$\omega$ was not chosen ``excessively small
with respect to $\exp(-\tau)$.}
and
$\varepsilon < \tilde \varepsilon < \varepsilon +
C \exp(-\tau)$ (the same constant $C$)
having the following properties:\roster
\item It is Lipschitz continuous in the sense that
whenever both $H_1,H_1'$ are from (2.27)
such that $H_1 - H_1'$ is from a class
$\Cal H^{\varepsilon', \omega}$, $\varepsilon' < \varepsilon$,
then $\tilde H_1 - \tilde H_1'$ is from
$\Cal H^{\varepsilon'', \omega}$
where
$\varepsilon'' = \varepsilon' \exp(-C'\tau) $
with another constant $C' = C'(\nu) $.
\item If $y$ is a ground state of $H_G +H_1 +\tilde H_1$, i.e. if it
minimizes the sum $\sum_t \tilde h_t(y)$, where $\tilde h$ denotes
the density of free energy of the model with
Hamiltonian $H_G +H_1 +
\tilde H_1$,
then the stratified $y$--like Gibbs state of the
Hamiltonian $H= H_G +H_1 +H_P$ exists. \footnote{Ground state of
$H_G +H_1 +\tilde H_1$, not of $ H$ resp. $H_G + H_1$! And we are looking for Gibbs states of $H$.}
\item In particular, if we choose
$\varepsilon' $ such that $\varepsilon - \varepsilon' >
C \exp(-\tau)$
then there is
\footnote{ By the inversion mapping theorem.}
for any $\tilde H \in
\Cal H^{\varepsilon',\omega}$ a
preimage $ H = H_G +H_P +H_1, \ H_1 \in
\Cal H^{\varepsilon,
\omega}$ such that $H_1 +\tilde H_1 = \tilde H$
and such that there is one to one
correspondence between the stratified ground states
of
$H_G +\tilde H$ and the stratified Gibbs states
\footnote { More precisely we mean here
only the stratified Gibbs states {\it constructed
by our Main Theorem\/}. However, we do not expect an existence
of any other
stratified Gibbs states of $H$.}
of the Hamiltonian $H$.
\endroster
\endproclaim
\remark{Notes on the proof}
See section 8 and also \cite{EMZ} for more information.
The interpretation of the
additional interaction $\tilde H_1$ in (2.29)
is straightforward if we start with a
abstract \ps \ model ($H_1 = 0$) in (2.27). Then the cubic interaction $\Phi_{\square}$ for the
new hamiltonian $ \tilde H_1$ arises very naturally; namely
it suffices
to perform a partial summation, over cubes, in the formulas
$$\log Z^y_{\updownarrow}(\Lambda)= -\sum_{t \in \Lambda}
e_t(y) +\sum_T k_T(y) \ \tag 2.30$$
and then to use (3.1). We actually have
$k_T =n_T \prod_i \exp(-F(\gb_i)$) with some
combinatorial coefficients $n_T$, and the sum is over all clusters
$T =\{\gb_i\}$ of ``recolorable'' contours with the external colour $y \in
\es$.
The partial summation is taken
over all the cluster expansion terms $k_T(y)$
which can be ``packed'' into a given cube $\square$ but not into
a smaller cube.
If we start with a nontrivial $H^{\omega,\varepsilon}$
then the situation is
analogous;
see section 8 and \cite{EMZ}
for some arguments; the aplication of Corollary above
to {\it spin\/} models (not only to models
already formulated in the canonical
form (2.12)) requires some additional discussion.
\endremark
\head III. The Concept of a Mixed (Partially Expanded) Model. Recoloring.
\endhead
\vskip1mm
This is the final and the main part of the paper. It is devoted to the
construction of suitable
{\it expansions\/} of partition functions of models considered in part II.
Here we introduce the important technical notion of a ``mixed'' (or, partially
expanded) model which serves as an {\it intermediate construction\/}
between the original concept of an ``abstract Pirogov -- Sinai model''
and our final aim which is an utmost expansion of the partition functions
of the considered model.
Cluster expansions were always an important tool
in the Pirogov -- Sinai theory. However, in previous versions of this
theory,
the
expansions were viewed merely as
some auxiliary technique applied to the
study of special polymer models (contour models) which were constructed first.
One could think that the cluster expansion method could be replaced by
``something else'' giving ``comparably nice'' expressions (or, possibly,
suitable
bounds only) for the partition functions of the contour models.
This is not so here where the idea of a partial expansion enters
even our basic terminology, namely the concept of the mixed model.
For example an analogy of the important notion of a ``metastable
model'' (see [Z]) can {\it not\/} be apparently even defined here
without the
language of expansions; also the
very formulation of our Main Theorem is based on this language.
Of course the idea of a ``partial expansion'' is not at all new.
It was used (in various context also in situations
close to the subject of the presented paper - see
\cite{I}, \cite{HKZ}, \cite{B},\dots by many authors
mostly as an important but auxiliary tool, while in our
formulation it is really the cornerstone of our theory.
Our basic expansion step (Theorem 3, Theorems 4 and 5)
-- called recoloring
by us {\it incorporates\/} some of the usual cluster expansion ideology
(based on the expansion by power series)
into the very construction of the contour functional.
As a consequence, our use of cluster expansion
technique is {\it selfcontained\/} and we need no references
to the literature. We can, however, mention \cite{M}, \cite{KP}, \cite{DZ}
(as the papers having direct influence on the present paper)
from the numerous literature on the subject of cluster expansions.
The construction of one ``recoloring step'' (Theorem 4 and
Theorem 5) \footnote
{In fact, one could formulate directly a version of Theorem
4 recoloring all the shifts of $\gb$ at once
i.e. giving directly Theorem 5. In future expositions
of our method we plan to follow this more direct
approach.}
will not yet give the required
expansion of the model. This step must be {\it repeated\/}
many times (infinitely many in the thermodynamic limit).
The iterative
nature of our constructions
cannot be hidden ``somewhere into the proofs''
but appears already at the level of the basic notions.
We organize this part of the paper as follows.
In section 1 we analyze the notion of a ``cluster''
(of supports of contours or, more generally, of another clusters);
the clusters are then identified as suitable trees
mapped to $\zv$.
Then, in section 2, we define the central concept of a {\it mixed\/}
model.
This notion corresponds to an idea of a ``partially
expanded model''; however it is useful to consider such a concept
in a broader sense.
Section 3 describes an important procedure
-- the``recoloring'' (i.e. the removal of $\gb$ from the model $\&$ adjustment
of the new cluster series such that the partition functions would not
change) of an interior contour $\gb$, in the context of a general
mixed model. The important concept of a recolorable contour
is introduced here: it corresponds, roughly
speaking, to
the validity of the Peierls condition for the contour
functional $ F(\gb)$.
Section 4 applies the result of section 3 in such a way that
recoloring of all the shifts of $\gb$ is obtained; the resulting
new mixed model is again a horizontally
translation invariant one if the original
mixed model satisfied this property.
Technically, sections 3 and 4 (complemented by
later sections 6 and 7)
form the core of our paper.
Later sections 5, 6, 7 are then devoted to the problems of
the succesive construction
of ``more expanded'' mixed models:
An important intermediate result
is Theorem 6 (section 6) giving a sufficient condition
for the recolorability of an interior contour system
in a general
mixed model.
Namely, to have more specific examples of recolorable
contour systems we
introduce there
a related but
better controllable notion of a {\it small\/},
more precisely {\it extremally small\/}
contour system which is more useful than a (more general)
notion of a recolorable contour system.
The message of the sections 5 to 7 is roughly speaking the following:
once there are some small contour systems in the mixed model then
there is still ``something left to recolor'', i.e
there are still
some recolorable interior contour systems in the model.
The notion of an extremally small contour system is an
elaboration of the older idea of a ``small'' or ``stable'' contour
(\cite {Z}). Notice that small resp. extremally small contour systems
can contain some
``large'' (``not extremally small'') contour systems
as their ``interior'' contour subsystems.
Theorem 6 is proved with the help of Theorem 7; the latter is
actually some general statement about the ``connectivity constant''
of some special (``tight'') sets appearing in the
study of extremally small contour systems.
Only after finishing the sequence
of all the expansions (recolorings) organized by Theorem 5
we will be able to say what the {\it metastable\/} model
is -- in Section 8. This will be defined as a submodel
of the original abstract \ps\ model where only those configurations
containing {\it no\/} ``residual \footnote{
``Residual'' means surviving in the ``totally expanded'' model:
by the totally expanded model we mean the final mixed model remaining at the moment
when the inductive procedure of its partial expansions was
completed. See Section 8 below.}
external contours'' will be admitted!
Section 8
formulates then our main result, using the
quantities called ``metastable free energies'' just constructed
by expansions. It turns out that that
the minimality of the metastable
free energy of some $y \in \es$ really means that there are {\it no contours
at all\/} in the totally expanded model under such a boundary condition
i.e. the metastable model corresponding to $y$ gives an appropriate
$y $-- th {\it Gibbs state\/}.
The fact that under ``stable''
boundary conditions, ``everything is recolorable''
(i. e. a complete expansion of the partition functions is obtained)
is the
core of the proof of Main Theorem. Having proved the
preparatory Theorems 5, 6 this will be almost
a tautology.
Our new method based on Theorem 5 and Theorem 6
replaces the previous coarser arguments
from \cite{Z} which cannot be used in these new situations.
However, even in the situation
of \cite{Z} our new method is simpler
(at least conceptionally) and more powerful:
the {\it bounds\/} for the partition functions employed in \cite{Z}
are now systematically
replaced by statements on {\it quick convergence
of the corresponding expansions\/}. See \cite{ZRO}.
We plan to show the advantages of this new approach in the study of further
situations which are not covered by the usual variants of
the Pirogov -- Sinai theory.
\head 1. Clusters
\endhead
\vskip1mm
This section prepares some technical notions and constructions
needed for the proper formulation of the expansions which are used below.
Cluster expansions of partition functions of polymer models are
usually written, in the literature, in the following
form:
$$ \log Z^y(\Lambda ) =- \sum_{t \in \Lambda} e_t(y)
+\sum _{T \subset \Lambda} k_{T}(y). \tag 3.0$$
Actually, we will be interested here in the expansions
of the partition functions $\log Z^y_{\updownarrow}
(\Lambda)$ where $y \in \es$ and
the condition $T \subset \Lambda$ will
have
a more restrictive
meaning, see below in (3.20).
\footnote{ We will have always
$\dist (S,\Lambda^c) \geq \diam S$ for
any set $S $ used in the construction of
the cluster $T$.}
The quantities
\ $ k_T(y), \ T \subset \zv$, are some local functions
\footnote{ Actually, they will be given as products of
exponentials of contour functionals, like in
(2.26). See the forthcoming sections for the extensive
discussion of the values $k_T$.}
dependent on $T$ and on the ``local colour''$y$
(in fact on
$y_{\vvt}$, see below). They are
indexed by ``connected''(in some generalized sense)
clusters $T$ (see below for more
details about this notion) and they are ``quickly decaying''
e.g. like (this will be the form used by us)
$$ | k_{T}(y)| \leq \varepsilon^{\conn T}
\tag 3.1 $$
where $ \conn T $ is something like the ``cardinality of a
minimal commensurately connected
set containing the cluster $ T $''.
See Definition 4 below for
the definition
of the quantity $\conn T $ which will be used in our
later considerations.
The constant $\varepsilon > 0$ will be asssumed
to be {\it sufficiently small\/}, often it will
be of the order $\varepsilon = \exp (- C\tau)$
for a suitable constant $C$,
This will
be our {\it final goal\/}: establishing of such expansions for
a collection -- as large as possible -- of diluted partition
functions of the given model.
More complete information will say that the quantities $ k_{T} $
are in fact sums of quantities indexed by some
``clusters of sets (resp. of contours) ''
(and having a value which is a $\pm$ product, over the cluster, of
contour functionals $\exp(-F(\gb_i))$) having the given support $ T $.
While one can ignore the detailed description of the
structure of $k_T$ when applying the above expressions
(e.g. in order to obtain useful {\it bounds\/} for partition functions
-- and this was the typical application of the cluster expansions in
most previous variants of the \ps \ theory) here it
will be necessary to retain the more precise information because
these expansions will be iterated repeatedly many times.
Before defining the notion of a cluster formally, we start
with the explanation of the notion of $ \conn T $
for the case when $ T $ is a {\it set\/}. Our definition relates such a
notion
to finding of some shortest ``commensurately connected'' superset containing
$T$;
this will be important later in this section when analogous
construction will be applied also to a general cluster.
We start in fact with the definition of a closely
related
quantity
denoted by $ \Conn T $ which is defined in a more direct
way.
In the definition of $\conn T$ (see below)
we will use the notion of an abstract {\it tree\/},
sometimes also with a specified {\it root\/}:
\definition{Abstract trees} An abstract tree
is defined as an equivalence class, with respect to isomorphisms of graphs,
of nonoriented (binary) graphs without cycles. (By a cycle of a graph $G$
we mean a collection $\{\{t_1,t_2\},\dots,\{t_n,t_1\}\}, t_n \ne t_2$
composed of bonds of $G$.)
If we wish to specify also the {\it root\/} of such a tree
(i.e. mark one vertex of the graph),
then such a tree can be concisely desribed also
in a {\it recursive\/} way, just by
specifying the collection of all subtrees, with marked roots,
emerging if the root of the tree
is removed.
\enddefinition
\remark {Note}
Below, an identification of a cluster with a suitable tree
will be given. This
suggests that the following idea of the summation
over cluster expansion series will be developed below:
Summation over clusters will be reformulated
as summation over trees,
and instead of estimating the number of various clusters with the same
length we will rather employ the idea of the
{\it recursive\/} summation over all
trees, based on the successive summation over the {\it
outer\/}
bonds of considered trees.
It seems that this method gives good estimates
and it also offers possibilities of a generalization
to other interesting models.
Therefore, we are developing this method here,
in spite of the fact that the treatment given below is maybe somewhere
too much detailed for the purposes of the forthcoming text.
Namely, a weaker version could be made, which would be
close in its spirit to
our later approach of section 7 (based on the notion of
a tight set;
see the proof of Theorem 7).
Nevertheless, we follow the method of summation over
trees here,
considering it also as a suitable
reference for possible further applications
of the method. (A summation over trees
is used in the paper [CONZ] which is under preparation.)
\endremark
\definition{Commensurate trees on $\zv$}
By a {\it commensurate tree $\Cal T$ on $\zv$\/}
we will mean the object satisfying the following
three requirements:
\roster
\item It is a pair $\Cal T = (G,\phi)$
consisting of {\it an abstract tree \/} $G$
and a {\it mapping\/} $\phi$ of this abstract tree $G$ to $\zv$.
Such a mapping
can be constructed, after fixing the root of the given abstract tree,
in a recursive way (following the recursive definition
of an abstract tree above): simply the image of the newly added root
is specified at each stage of the recursive construction.
The vertices of the
abstract tree $G$
are mapped (generally
not one to one) onto some subset of $\zv$ which will be called the
{\it support\/} of the given commensurate tree and
denoted by $\supp \Cal T$.
Notice that several
vertices of the
given abstract tree $G$ can be mapped to the same
$t\in\zv$.
\item The {\it bonds\/} of the tree $\Cal T$
(more precisely, the corresponding images of bonds of $G$
in $\zv$)
constructed in (1) are always (unordered)
pairs of the following special type:
$$ \{t, s\}\, ; s = t + 2^k\vec e_i $$
where $ k\in\en\, , t \in 2^k \zv\, $ and where $\, \vec e_i\, $
is either zero
or a vector of the canonical base of $\zv$.
More precisely we consider all the bonds
$\{t=\phi(A), s=\phi(B)\}$
which are images under $\phi$ of the corresponding
bonds $\{A,B\}$ of the abstract tree $G$.
(We put no limitations on the number of such bonds per a given
pair $\{t,s\}$.)
\item The {\it commensurability\/} of $\Cal T$
is meant in the following sense: if $\{\phi(A),
\phi(B)\}$ and \
$\{ \phi(A), \phi(C)\}$ are two neighboring
bonds of the tree $\Cal T$ then
$$ {1 \over 2} \rho(\phi(A),\phi(B)) \leq \rho(\phi(A),
\phi(C)) \leq 2 \rho(\phi(A),\phi( B))\, \,
\tag 3.2 $$
where the distance $\rho(\phi(A),\phi(B))$
between $\phi(A) =t $ and $\phi(B) =s $ is defined
as $2^k$ resp $1$ according to
whether $s=t+2^k \vec e_i$ or $s=t$
in the
above relation.
\item We define the length of such a tree
as the total number of its bonds {\it excluding\/}
all the bonds (``loops'') of the type
$\{\phi(A) = t, \phi(B) =t\}$.
\endroster
\enddefinition
\remark{Note}
The usage of the lattices $2^k\zv$ and our very notion of
a commensurability will be quite important in the following. The choice
of the factor $2$ in (3.2) is more or less arbitrary but convenient later.
We should notice that later,
in the proof of Theorem 6 below, the notion of commensurability will
be transcripted to an {\it alternate\/} language based on the usage of the
unit {\it cubes\/} from
lattices $2^k \zv$ (considered as cubes from the original lattice
$\zv$) instead of the
employment of the {\it bonds\/} of the type above.
\endremark
\definition{The quantities $ \operatorname{Conn T}$ and $ \operatorname{conn T}$}
Given any set $T\subset\zv$ we assign to it a
shortest possible commensurate tree containing
for any $t\in T$ {\it at least one bond\/} of the type $\{t, t\}$.
We will denote one such tree (it is often not determined uniquely, even
if its root is already selected) as $\Cal T=
\Cal T(T) \, $.
We recall that
the length of the tree was defined by (4) above and therefore
the ``loops'' (bond of the type $\{t,t\}$) are not contributing to the length of the
tree; the condition that all such loops are in the considered tree
can be replaced by requiring that
any $t\in T$ belongs to some bond $\{A,B\}$
of $\Cal T$
having the
length $\rho(A,B) \leq 2$.
Define the auxiliary quantity $\Conn T$ as the
{\it length\/} (see the point (4) above)
of the above tree
$\Cal T(T)$.
In the following, it will be more useful to have a
modified version of this quantity, denoted by $\conn T$ and defined as
follows:
$$
\conn T = \Conn T +[3 \nu\log_2\diam T] + 6 \nu
\tag 3.4 $$
\enddefinition
\remark{Note}
(3.4) will be a more adequate quantity than $\Conn T$ in what
follows; see Proposition below. Namely, the clusters of sets
will be defined below in a recursive way as collections of objects
(sets or contour systems) having the property that
their diameter is not smaller than their distance
to some other (bigger) object of the collection''; and
when constructing additional commensurate path connecting
a given set $T$ with a point in distance $\diam T$
one requires an additional amount of \ $\approx \log_2
\diam T$ steps:
\endremark
\proclaim {Lemma 1}
Let $\rho(t, s) = d$. Then there is a commensurate {\it path\/}
starting by the bond of the type $\{t, t\}$ and ending
by $\{s, s\}$
having the length at most
$[\,3 \nu \ log_2 \ d\, ]$.
\endproclaim
\demo{Proof of Lemma 1}
It follows easily from the following considerations: first notice that it suffices
to consider the case
of the dimension $\nu = 1$. Consider now the path on $\zet$ with steps
having the lengths (we start and end with loops but we do not count them,
$k \geq 1$)
$$ 1, 2, 4, \dots, 2^k, \dots, 4, 2, 1$$
which overcomes the distance $d= 3\cdot 2^k -4$. The length of this path is
$2 k - 1 $. If
$$ 3 \cdot 2^k - 4< d' <5 \cdot 2^k -4$$
then it is possible to construct a commensurate path overcoming the
distance $d'$ simply by {\it doubling\/} some
of the
steps in the first half (including the middle step) of this sequence.
Thus we need at most $2 k - 1 + k+1 \, \leq\, 3 \,[\log_2 d'] \, $ steps
to overcome any distance $d'$ from the interval
$[2^k,2^{k+1})$ (even from the interval $[0, \ 5 \cdot 2^k
-4)$) which completes the proof
for $\nu =1 $. The multidimensional case
is analogous, just construct $\nu$ paths in each coordinate axis
and intertwine their steps together suitably.
\enddemo
Now we come to the definition of a {\it cluster\/}.
The notion of a cluster of subsets \footnote{ Actually,
only {\it some
\/} subsets of $\zv$
will be employed in the construction of
a cluster. Namely the supports of so called
``recolorable'' contours or contour systems,
see below in sections 3 -- 5.} of $\zv$
is defined recursively, retaining the
letter T for the notation of clusters, as follows:
\definition{Clusters of sets $\supp \gb$}
\roster
\item "{i)}" Any set
$$ T = \supp \gb$$
where $\gb$ is a contour or
a contour system (``recolorable'' one, see below)
is a cluster.
\item "ii)" If $T_i$ are some clusters, $T_i \ne T_j$
for $i \ne j$ and
$T_0$ is from i) such that
\footnote{We will consider in what follows only the
``standard'' clusters (see below). Then we
will have actually a stronger condition
$\supp T_i \cap \cap (\supp T_0)^c \ne \emptyset$;
see section 3 of part III.}
$$ \dist ( T_0, \supp T_i) \leq 4 \ \text{min}\ \{ \diam(\supp T_0),
\diam
(\supp T_i)\}
\tag 3.5 $$
holds for each $i\geq 1$, then the pair
$ T= (T _0, \{T_i\})$
is again a cluster. We denote by
$ \supp T = T_0 \cup \cup_i \supp T_i\,\,$.
The set $T_0$ will be called the {\it core\/}
\footnote{The fact that we have {\it one\/} core
and not, say, multiple cores consisting of several
horizontal shifts of the same $T_0$ (which
is related to the fact that we employ
the ordering $\prec$ and not $\prec \prec$
in the definition of a cluster) is related to our
method, based on the ``lexicographical''
Theorems
3 and 4. Clusters with multiple cores
(and no need for the lexicographical order) would appear
if these theorems would be reformulated for
{\it several\/} copies of $\gb$ {\it at once\/}.
We plan to
prefer such an approach in the future.}
of $T$.
The external colour, let us denote it by $y$,
of $\gb$ will be called the
{\it colour\/} of the cluster $T$ and we will assume
that $T_i$ already have the {\it same\/} colour
$y \in \es$ \footnote{Of course there is, as always,
an ambiquity
in the extension of $y$ to the whole $\zv$.}.
\endroster
\enddefinition
\remark{Note}
The condition (3.5) is a technical one; its adequateness
(with respect to our actual constructions) will be seen later
in Theorem 4. The additional ``logdiam'' term added in
the definition of $\conn T$
(in comparison to $\Conn T$)
will be seen to be related to our very
formulation of the condition
(3.5). See Proposition below.
\definition{A tree associated to a cluster}
We assign, to any cluster $T$, a commensurate
tree $\Cal T$ defined as follows:\ If
the trees \ $\Cal T_0$ and $\Cal T_i$ were
already constructed by definition above
and by the induction assumption for $T_i$ \ (recall that $\Conn T_0 =
|\Cal T_0| $ where $\Cal T_0$ is a shortest
commensurately connected tree whose support contains
$T_0$)
then we define $\Cal T$ as the shortest possible
commensurately connected tree containing, as mutually disjoint
subtrees (with disjoint bonds $\{A,B\}$)
all the trees \
$\Cal T_0$ and \ $\Cal T_i$
and such that all the components
of \ $\Cal T \setminus \Cal T_0$ start with
some loop of the type $\{t,t\}, t \in T_0$.
To have an idea about the length of $\Cal T$ consider
the following particular construction of a tree
whose length should be ``close'' to that of $\Cal T$.
Moreover one has a clearer idea about its shape, see
below in (3.6)).
$$ \Cal T' =\Cal T_0\, \cup \, \cup_i (\Cal T_i \cup P_i)$$
where $P_i$ are some shortest possible commensurate
paths,
starting
in some loop $\{s_i, s'_i\}$ of $\Cal T_0$ and ending in some loop
$\{t_i, t'_i\}$ of $\Cal T_i$.
In analogy to the case of a set $T$, the quantity $\Conn T$
is now defined as the length of the tree $\Cal T $ and we
put (compare (3.4))
$$ \conn T = \Conn T + [ 3 \nu \log_2 \diam \supp T] +
6\nu .
\tag 3.4'
$$
\enddefinition
In order to reconstruct back the original cluster $T$ from a given tree
$\Cal T$
the following notion of a {\it standard cluster\/}
will be useful:
\definition{Partial ordering of finite subsets of $\zv$}
Fix the partial ordering $\prec$
on the collection $\Cal F$ of all finite
subsets of $\zv$
resp. (this will be used later, starting from
Theorem 5)
the partial ordering $\Cal A \prec \prec \Cal B$
between the equivalence classes
$\Cal A =\{A' = A + t, t \in \zv\}$
(the ``factorization of $\prec$
with respect to all shifts
in $\zv$'')
given by the following
requirements: The ordering $\prec \prec$
extends the partial ordering
$\diam A < \diam B$
(recall that we use the $l_{\infty}$ norm on $\zv$
everywhere)
and it is invariant
with respect to all shifts in $\zv$. The ordering
$\prec$ moreover extends
the
ordering $A \prec A'$ iff
$ A'= A -t$, $t \prec_l
0$
where $\prec_l$
denotes the {\it lexicographical ordering\/} on $\zv$.
\enddefinition
\definition{Total ordering}
One can, and will
extend the partial ordering
$\prec $ (analogously,
the partial ordering $\prec \prec$)
to a {\it total\/} ordering on
$\Cal F$ (resp. on its factorization
w.r. to all shifts). \enddefinition
\remark{Note} There is, of course, still some arbitrariness in such an
extension. For example, one can add, to the requirements
1) and 2),
also the following two requirements: 3) $A \subsetneq \tilde B \Rightarrow A \prec B$,
where $\tilde B$ denotes again a suitable shift of $B$
and even, say,
\footnote{ This will be convenient later, when defining
{\it external contours\/} (in section 9), though this
notion will be only
of a marginal importance to us.} another relation \ \
4) $ \vva \subsetneq V_{\updownarrow}(\tilde B)
\Rightarrow A \prec \prec B $
if the former is valid for {\it some\/}
shift
$\tilde B = B +t$
and if we denote by $ \vva = \{ t \in \zv:\ \dist(t,A)
\leq \diam A\} $. (This temporary notation,
used only here should not
be confused
with the notion of $\vvd = \cup_i \vvi$
defined before for $\Cal D =\{\gb_i\}$.)
This is natural
and {\it consistent\/}, as an inspection shows,
with
the requirements on $\prec$ resp. $\prec \prec$
given above. We can say that
$A \prec \prec B$ holds
if
at least {\it one\/} of the
relations
$ A \subset \tilde B, \diam A \leq \diam B,
\vva \subset V_{\updownarrow}(\tilde B)$
($\tilde B$ is suitable shift of $B$) is not equal to $=$.
\endremark
Before coming to the following definition, let us fix
the {\it total\/} ordering $\prec$ defined above.
Note that while the choice of the extension of $\prec$
played
no role in the definition of
contours and their admissibility,
the fact that we have a total ordering
{\it will\/} be important now.
\definition{Standard clusters}
Say
that a cluster $ T= (T _0, \{T_i\})$ is a {\it standard\/} one
if $S \prec T_0$ holds for any set $S$ used (as a core)
in the recurrent
definition of $T_i$.
\enddefinition
Notice that since $\Conn T $ is not greater than $|\Cal T'|$ we have
the inequality
$$ \conn T \leq \Conn T_0 + \sum_i \Conn T_i + \sum_i l_i +
[ 3 \nu\log_2 \diam \supp T ] + 6\nu \tag 3.6$$
where $l_i$ are the lengths of the
paths $P_i$ used in the definition of $\Cal T'$.
\footnote{
In fact, in Theorem 6 we will show that for all clusters considered later
by us, the quantities $\conn T$ and $|\supp T|$ will be of the
{\it same order\/}. Moreover, one could apparently
rewrite the present section for
this (narrower) setting in the spirit analogous to that of
later Theorem 6,
without employing the bothering (but small !)
logdiam terms. We prefer the more general exposition here in view
of a wider applicability of the estimates obtained here also to other
situations.}
The following estimate will be used later in Theorems 4
and 5 (though in
a slightly changed form). It says that having established a
slightly stronger version of the estimate (3.1) for {\it sets\/} $ T$
one obtains (3.1) also for all {\it clusters\/} $T$ if the quantities $k_T$
are given by the recurrent formulas below.
The quantity $\Conn T$
does not seem to have comparably nice
properties; the additional `` logdiam'' term in
our definition of $\conn T$ seems to be essential for our ability
to give a recurrent proof of (3.9).
Next we formulate two auxiliary results: The first one will be
directly used later (in a slightly different form not changing its
essence -- see the proof of Theorem 5). On the other hand, the second result
is its corollary which we formulate in a {\it more general\/} setting
(Theorem 2)
which will be possibly interesting
also in other situations where
our method can be applied. This latter result
resembles the classical Mayer method
(see \cite{R1}).
\proclaim{Proposition 1}
Assume that the quantities $k_T$ are defined recursively by formulas
$$ k_T = k_{T_0} \prod_i k_{T_i} \ \
\text{whenever} \ \ T =(T_0,\{T_i\}). \tag 3.7 $$
Assume that for the {\it sets\/} $T_0 \subset \zv$ the following stronger variant of (3.1)
is valid:
$$ | k_{T_0} | \leq \varepsilon ^{\Conn T_0 + 6 \nu(\log_2 (\diam T_0 + 4)+2)}
. \tag 3.8 $$
Then the estimate
$$ |k_T| \leq \varepsilon ^{\conn T} \tag 3.9 $$
holds also for all the clusters $ T = \{T_0,\{T_i\}\}$, with $\conn
T$ defined by the preceding definition, assuming that it is already
valid for all clusters $T_i \ ;\ i \ne 0$, in (3.7).
\endproclaim
\demo{Proof}
We have to prove that
$$ \Conn T_0 + 6 \nu \log_2 (\diam T_0 +4)+12\nu + \sum_i \conn T_i \geq \conn T
. \tag 3.10 $$
Notice first the following simple estimate. Define the support $\supp T$
and
the diameter
$\diam T$ of a cluster $T=(T_0,\{T_i\})$ recursively by
putting $\supp T = \supp T_0 \cup \supp T_i$
and $\diam T = \diam \supp T$.
\proclaim{Lemma 2}
Let $T_j$ be the longest of all clusters $T_i$
(maximizing the diameter). Then
$$ \log_2 \diam T \leq \log_2 (\diam T_0 +4) +
\log_2\diam T_j.$$
\endproclaim
\demo{Proof of Lemma 2}
This is straightforward: notice that the condition
(3.5) implies the bound
$$ \diam T \leq \diam T_0 + (1+1+1+1) \diam T_j . $$
Then we use the inequality $\log_2 (x + 4y) \leq \log_2 (x+4) + \log_2 y$
which is surely valid for $ x \geq 1 \text{ and }
y \geq 1$.
\enddemo
The idea of the proof of (3.10) now
is to reduce it to (3.10'') below, and to ``feed'' the
necessary increment of
$\conn T -\sum_i \conn T_i -\Conn T_0$ by the
``superfluous
logdiam
term'' in (3.8) resp. by the value $\conn T_j -\Conn T_j$
of the longest cluster $T_j$
(if it is much bigger than $T_0$).
Notice that it suffices to establish
the following bound, from which the required bound (3.10)
is obtained
by summing it with $3\times$
the bound of Lemma 2.
$$ \Conn T_0 + 3 \nu\log_2 (\diam T_0 +4) + 6 \nu +
\sum_{i \neq j} \conn T_i
+ \Conn T_j \geq \Conn T \tag 3.10' $$
Really, this is surely valid because
we can rewrite it (notice that
$l_j \leq 3 \nu\log_2 (\diam T_0 ) + 6\nu$
by Lemma 1 !) in a stronger form
$$ \Conn T_0 + \sum_i (\Conn T_i + l_i) \geq \Conn T
\tag 3.10''
$$
where $l_i \leq \conn T_i - \Conn T_i$ denotes the length of the path $P_i$ used in the
definition of the auxiliary tree $\Cal T'$ (see Definition 4).
Namely, the bound
$ l_j \leq 3 \nu \log_2 \diam T_0 +6\nu$ \ follows
from the condition (3.5)
and Lemma 1.
The validity of the last inequality (3.10'')
follows from the very definition
of $\Conn T$ (see (3.4), (3.6)) and this completes the proof
of Proposition 1.
\enddemo
\definition{Notations}
In the following we will usually write, for clusters $T$,
$ t\in T, T\subset \Lambda,
\dots$
instead of the more precise notations
$t\in \supp T, \supp T \subset \Lambda,
\dots $.
By writing $G \in T$ we will mark
the situation when the set $G=\supp \gb $ was used in the
recursive definition of $T$
(as the ``core'' of some intermediate cluster used during the construction)
of the cluster $T$. We will also
extend the notation $\sqsubset$,
more generally $\subset\subset$ to {\it clusters\/}
(compare (2.19), but here we have a cluster
instead of a single contour system)
$$ T \ssubset \Lambda \ \ \text{iff} \ \ G \subset
\subset
\Lambda \
\text{for each}\
G \in T. \tag 3.11$$
\enddefinition
Finally we have (see below for the proof)
one important
consequence of the condition (3.1), to be
used later in the formulation of our main result.
\proclaim {Proposition 2}
If there is a small $\varepsilon$ such that for each cluster $T$
and each $y \in \es$,
$$ |k_T(y)| \leq \varepsilon^{\conn T} $$
then the cluster
series with the terms $k_T$ quickly converge in the following sense:
for any
$t \in \zet$, any $y \in \es$ and for any $d \in \en$ we have
$$ \sum_{T: \ t \in T \& \ \Conn T
\geq d} |k_T(y)|
\leq (C\varepsilon)^d \tag 3.12 $$
and analogously for the condition $\conn T \geq d$ replaced
(e.g.) by $|\supp T| \geq d$.
\endproclaim
We require such a bound only for the
(``more natural'')
quantity $\Conn T$. Recall that our introduction of the quantity $\conn T$
was
motivated by the necessity to derive (3.1) for all {\it clusters}
from something like
(3.8) which should be assumed to be valid for all {\it sets} $T$. Once
we {\it have\/} (3.1) for {\it all clusters\/} we can forget the
quantity $\conn T$ and replace it by $\Conn T$ if the convergence
of the cluster expansion is investigated.
We will prove Proposition 2 in a broader setting,
related to usual estimates in the theory of the
Mayer expansions (see the book [R1]).
It is easy to understand that Theorem 2 below
actually {\it generalizes\/} Proposition 2 above:
Below we identify any cluster $T$ with some
commensurate tree
$\Cal T$ on $\zv$. The number of standard
clusters corresponding
to $\Cal T$ will be shown to be at most $6^{|\Cal T|}$.
Having this in mind the
forthcoming Theorem 2 can be formulated for
quantities $ k_{\Cal T}$ indexed by
commensurate {\it trees\/} $\Cal T$ on $\zv$
instead of clusters.
\definition{Reconstruction of the cluster $T$
from $\Cal T (T)$} \enddefinition
The mapping from clusters to trees constructed above
is not one to one.
First notice that given a ``connecting
commensurate tree'' $\Cal T$ on a {\it set\/}
of cardinality $n$,
there are no more than $2^n$ possibilities how to
recognize
the original set $T$ (if there is some)
just by specifying the points from the support of
$\Cal T$ not belonging to $T$. Second, assume
that clusters are already constructed
(recurrently)
from {\it trees\/} instead of sets.
How many ways are there then to
reconstruct the trees $\Cal T_0$ etc. from $\Cal T$ ?
Assign to any bond of $\Cal T$
a ``colour'' red, blue or grey
such that grey subtrees are interpreted
as the ``building blocks'' $T_0$ and $ T_i$
while red bonds denote the connecting subtrees --
except of the bonds ``ending the connecting
subtrees, going from $\Cal T_0$ to $\Cal T_i$''
($\Cal T_i$ were constructed in the
previous step of the recurrent definition of
$\Cal T$ ); these will be marked blue.
(Analogous coloring is used inside of each $\Cal T_i$,
by induction assumption.)
Look at the biggest (in $\prec$)
grey component; this must be
$\Cal T_0$
and analogously, taking red paths with blue ends
starting from $\Cal T_0$ one recognizes
the roots of the
subtrees corresponding to $\Cal T_i$ etc.
Thus, one has a bound (surely a very crude one !)
$6^n$
for the number
of standard
clusters $T$
with the same tree $\Cal T$ of the cardinality $n$.
In the forthcoming applications, {\it all\/}
the clusters constructed by us will be
standard ones and so we will not discuss the possible modifications of the
estimates given above which would be
needed if also nonstandard clusters would appear.
The following general result can be apparently useful
also in other situations
(see \cite{CONZ}); it is probably a ``folklore''
but we do not know a suitable reference.
\proclaim{Theorem 2}
Let the quantities $k_{\Cal T}$ be given as products of some quantities
denoted by $k_b$ or $k_{\{t,s\}}$ (see the commentary below)
\footnote{Recall that the ``bonds''
$\{A,B\}$ of an abstract tree $G$ are mapped by
$\phi$ to unoriented pairs of points
$\{t =\phi(A),s =\phi(B)\}$ from $\zv$.
The notation $k_b$ is used instead of a more explicit
notation $k_{\{t,s\}}$ for $b =\{t=\phi(A),s=\phi(B)\}$.}
$$ k_{\Cal T} = \prod_b k_{\,b} $$
where the product is
over all the ``bonds'' $b = \{\{\phi(A),\phi(B)\}\}$
of the commensurate tree $\Cal T =\{G,\phi\}$.
Assume that the quantities $k_b$ are nonnegative and $k_{\{t,t\}} = 1 $
for each $t$.
Let for any unordered pair $ b = \{t,s\} $
we have the estimate
$$\sum k_{\,b'} \leq q \tag
3.13 $$
where the summation in (3.13)
is over all unordered pairs $ b' =\{s,u\}, s \ne u $ which are
commensurate with $b$ and $q$
is some small, e.g. $q < 1/4$, positive constant.
Then for any pair $b$ we have the following bound
for the sum over all commensurate trees $\Cal T$
containing the ``bond'' $b$ as its ``extremal bond'' and
having the length
at least $2$:
$$ \sum k_{\Cal T} \leq k_b \cdot q' , \tag 3.14$$
where $q'$ can be chosen as $q' =3q$
(more precise estimate of $q'$ is given below). By the extremality
of \ $b = \{t =\phi(A),s =\phi(B)\}$
we mean that
one vertex of the pair $ \{A,B\}$
remains ``free'' (i.e. it is ``endvertex'' in the original abstract
tree.
\endproclaim
\remark{Note} One can optimize the value of $q'$. A better
value $q'_{\text{new}}$ of $q'$ can be sometimes
found
from the equation (see the end of the proof below)
$$ \exp( q'') = 1 +q'_{\text{new}} \tag 3.14'$$
where $q''$ denotes the supremum, over all bonds $b$,
of the sums on the left
hand side of (3.13)
but with modified terms
$$ k_{b'}''= { k_{b'}(1+q')\over 1- k_{b'}(1+q')} $$
and
where $q'$ is the previously established value
in (3.14).
One can further elaborate the process of finding the
optimal value
of $q'$. Namely, if all $k_b$ are small and if we have
already established
the smallness of $q'$ then we have the crude
bound, say $k_{b'}'' < 2 k_{b'} (1+q')$ and hence we have also the
inequality $q'' < 2q(1+q')$,
We will see below in the proof of theorem that
the solution $q'$ of the equation
$\exp(q'') = 1 + q'$ can be actually
placed into (3.14); it clearly
minorizes the value $q'_{\text{new}}$ above. See
below for more information on this argument.
\endremark
\demo{Proof} Apply the method of induction. Denote by
$\sum^{ 1$ different types of bonds
$b'$ stick to $b$; each of them has a multiplicity
$l \geq 1$ and each such bond $b'$ is the starting bond of some
(empty or nonempty) subtree, or $l$ subtrees. (Contributions of these situations
were just counted.) We get the final estimate
(notice that now we have the term $m!$ here)
$$ \sum^{ 0\,$
such that for any $T$,
$$
|k_T|\leq \varepsilon ^{\,\conn T} .
\tag 3.17
$$
\endroster
\remark{Notes}
{\bf 1.} We will not study too much
relations
between models in {\it different\/} volumes $\Lambda$.
In particular,
we will not look for analogies
of ``telescopic
equations'' usually formulated
for diluted partition functions
(see \cite{S}). We will {\it not\/} need this.
\newline
{\bf 2.} For all the configurations $(x,\{\gb_i\})$ considered below
(everywhere in what precedes our
Main Theorem, section 8) we will have
$x = x^{\text{best}}_{\{\gb_i\}}$ (see the discussion of this notion
at (2.9)).
In fact we consider the concept of a mixed model
as a generalization
\footnote{ It is important to notice that until section
8 we will assume {\it no fuctional dependence\/} of
the quantities $k_T(y)$ on the values $E(\gb)$
(and $e_t(y)$). Such dependence will
be studied only later, in the formulation of
Main Theorem,
when very concrete mixed
models will appear as partial expansions
of the original model.}
of the concept of
an abstract \ps \ model, contours $\gb$
being some ``abstract connected objects satisfying
the Peierls condition'' while spin configurations
will appear only as ``ground configurations''
$x^{\text{best}}$. Below we will usually omit the
superscript ``best''.
\newline
{\bf 3. }
We will later glue together, in our ``recoloring
procedure'', some contour systems $\gb$ and some clusters $T$ such that
$\dist(T, \supp\gb)\leq \diam T$; to form new clusters
of some new mixed model. This is one of the reasons
why we added, in the preceding section,
the ``safety constant'' $3 \nu \log_2 \diam T$
to the quantity $\Conn T$ in the definition of $\conn T$
in order
to keep the control over the connectivity properties
of the new clusters formed by such (recursive) procedure.
\newline
{\bf 4. }
The collection of allowed contours (more generally
of allowed
{\it configurations\/}, see below) may vary from
one mixed model to another. Typically the allowed set of contours
will be some
{\it subset\/} of the original collection
of contours (of some given abstract Pirogov -- Sinai model),
and this subset will become even {\it smaller\/}
when applying further expansions (recolorings) to the given model.
On the contrary, the collection of nonzero $k_T$ will
always {\it grow\/} with such an expansion. See the forthcoming section
for more details.
\newline
{\bf 5. }
The restriction of the assumption (3.16) to
nonzero products $k_T\, k_{UT}$ is related
to the fact that, in the forthcoming section, we will work, temporarily,
with
horizontally translation noninvariant models.
In fact the new cluster
quantities $k_T$ will be constructed successively in the lexicographic order,
through an infinite sequence of
intermediate (noninvariant) mixed models.
If the condition (3.16) is complemented by the assumption
that {\it both\/} $k_T$ and $k_{UT}$ are nonzero if at least {\it
one\/} of them is nonzero and if the
configuration space is horizontally invariant (in the sense of
what contour systems are allowed in $\zv$) we will speak about
the {\it translation invariant\/} mixed model.
\newline
{\bf 6. }
Having specified the {\it collection of allowed precontours\/}
of the mixed model
we do not require, in principle, that {\it all\/} admissible collections of allowed
precontours are allowed configurations of the mixed model.
We are quite general at this point and at the moment
we impose no special requirements on what collections of precontours
are really allowed in our model. See forthcoming sections 3, 4, 7
for a more concrete information on the actual choice
of the configuration space.
\endremark
The {\it partition function\/} of the mixed model
will be always the {\it strictly\/} diluted
one
:
$$
Z^{\alpha}
_{\updownarrow}
(\Lambda)=
\sum_{\Cal D \ssubset \Lambda} Z^{\alpha }_{\Cal D }
(\Lambda ) \tag 3.18
$$
where $\alpha $ is a boundary condition on $\partial_r
( \Lambda^c)$
(from $\es$) and $\Cal D $ is a contour system. The sum
is over all
contour systems $\Cal D$ such that $\Cal D \ssubset \Lambda$.
Recall that the notation
$\Cal D \ssubset
\Lambda $, analogously
$T\ssubset \Lambda$ means that the condition
$\gb \sqsubset \Lambda$ \ i.e.\
$\vv \cap \Lambda^c = \emptyset$
more generally $\gb \ssubset \Lambda$ for
a contour {\it system\/}
is satisfied for any contour $\gb$ of $\Cal D $
resp. any contour system $\gb \in T$.
Using the hamiltonian (3.15)
we define
$$
Z^{\alpha }_{\Cal D }(\Lambda )=
\exp(-\sum_i E(\gb_i))
\exp(-\sum_{t\in\Lambda }e_t(\delta ))
\exp(\sum_{T\ssubset \Lambda \setminus\supp\Cal D }k_T(\delta)) \tag 3.19
$$
where $\delta$, such that
$ \delta^{\text{ext}} = \alpha $ denotes the configuration
(see (2.9))
$\delta =(x_{ \Cal D })_{\supp \Cal D}^{\text{best}}$. Recall that it is stratified
outside of $ \supp \Cal D$.
\remark{Note}
It seems unnatural to use the symbol $Z$ \ for the ``mere Gibbs
factor'' (3.19). However, the case $k_T = 0$ is {\it not\/} the most
characteristic
example here. In a more general case, considered mixed model
corresponds actually to some partial {\it expansion\/} of the model (2.18).
Then (3.19) is really some partition function, corresponding to
an event ``$\Cal D$ is the collection of (still) nonexpanded
contours of the original model (2.18)''.
\endremark
\vskip1mm
\head 3. Recoloring of an internal contour system $\gb$
\endhead
This is a central construction of our approach,
{\it replacing\/} (together with the forthcoming constructions
of later sections)
the concept of a (metastable)
{\it contour model\/} used in the previous versions of the
Pirogov -- Sinai theory.
In this section we describe some
abstract, ``algebraic'' aspects of {\it one recoloring
step\/} (of an arbitrary mixed model). An invariant
(conserving the horizontal invariancy)
application of this construction will be given in the forthcoming section 4.
A suitable {\it sequence\/} of
(translation invariant) recoloring steps,
yielding
as its final result the ``total'' expansion of a given Pirogov -- Sinai
abstract model will be discussed later, starting from
Section 5.
Such a recoloring will be just one step towards the desired
``total expansion''
of the model, and this step is described in detail in
Theorem 3
and Theorem 4 below.
Roughly speaking, recoloring of a contour
(resp. of a contour system) $\gb$ will just mean a
{\it replacement\/} of a given mixed model by
another mixed model where $\gb$ will {\it not\/}
already be allowed as a contour (system) and where some {\it new\/}
quantities $k_T$ (for some {\it new} clusters $T$ containing
$G =\supp \gb$ as its core) will
appear. The remainder of the model will be kept intact
and the crucial fact will be that the diluted
partition functions (2.18) of both the original and recolored
model will be required to be the {\it same\/} for all finite volumes.
Let us start with the definition of the following
important quantity $A(\gb)$ which
``measures the instability of $x_{\gb}^{\ext}$ in
$V(\gb)$'' and which will play a key role later when defining
a rigorous substitute for (2.21);
see also the remark below.
There will be
several variants of the quantity $A(\gb)$ -- see below --
and the technical difference between
their definitions (essentially the decision what
volume will be used instead of $V(\gb)$)
will be quite important in this section, in spite
of the fact that the values of all these quantities will
be roughly the same.
Recall (it will be quite indispensable in what
follows) the definition of a ``protecting zone
over $V(\gb)$'' namely the notion
of $\vv$. Recall that
$ \vvd = \cup_i \vvi $. Recall the notations
for clusters $T$ and sets $\Lambda$:
$$ T \ccap \Lambda^c = \emptyset \ \ \Leftrightarrow\ \
T \ssubset
\Lambda \ \ \text{iff} \ \ \vvg \subset \Lambda
\ \ \text{for each} \ \
G =\supp \gb \in T
. \tag 3.20 $$
\definition{The quantity $A(\gb)$}
For any contour system $\gb$ denote by $A(\gb)$ the quantity
\footnote {More precisely $ A(\gb,x|{\vv})$. For
typhographical reasons, we
denote here the restriction
of $x$ to $V$ by
$x |V$ instead of the usual $ x_{V}$.
This configuration will be
always given from the context and this justifies the
shortened notation
$A(\gb)$. Notice also that the choice of the volume
(where the clusters $T$
live) in the sums above is somehow
arbitrary. See below (especially at (3.25)) for the
discussion of the other possible alternatives to $\vv$.
Finally, it is only a matter of arbitrariness
to require $T \ssubset $ and not $T \subset$ here.}
$$
A(\gb,x|\vv) =
\sum_{t\in V(\gb)}(e_t(y)- e_t(x)) +
\sum\Sb T\ssubset \vv\setminus \supp\gb \endSb
k_T(x)-
\sum_{T\ssubset \vv}k_T(y).
\tag 3.21
$$
Recall that here
$x=(x_{ \gb})^{\text{best}}$ is (see (2.9))
the configuration
minimizing the Hamiltonian $H(x)$ under the conditions $x_{\partial_r
\supp \gb} = \partial_r \gb \ (=
\{\gb_t, t \in \partial_r \supp \gb \})$, $x_{(V(\gb))^c}
= y$, $y=x_{\gb}^{\ext}$.
\enddefinition
\remark{Note}
Thus, to be able to determine $A(\gb)$
exactly we must know the
configuration $x$ on the whole volume $\vv$
because
$k_T(x)$ resp\. $k_T(y)$ depend on the values of $x$
resp\. $y$ on the set
$\cup_{G \in T} \vvg$.
However, the value of $x$ on $\vv$ will be normally
determined by the context in which the contour system
$\gb$ will appear and so we will often use the
shorter notation $A(\gb)$ instead of the more precise
notation $A(\gb,x|\vv)$ without any ambiguity.
\endremark
We will see that the quantity (see (2.11), (2.12)
for the definition of $E(\gb)$)
$$
F(\gb)=E(\gb)-A(\gb)
\tag 3.22
$$
is a useful exact substitute for the formal quantity
$F_{\text{formal}}$ from (2.21).
The choice of the set $\vv$ will guarantee (among other convenient
properties) that the
clusters constructed below having the ``core'' $\gb$
will be sufficiently
``tight''. (This would {\it not\/} be the case if we would take
mere set $V(\gb)$ there).
However, if $\gb$ is an ``interior contour system'' of some bigger
admissible collection $\gb\&\Cal D $ of all precontours of
some configuration $(x_{\Lambda }, \gb\&\Cal D )$ in a finite
volume $\Lambda$, the following modifications of the
quantity $A(\gb)$ could be considered too
and we mention them for comparison and better understanding
of the nature of $A(\gb)$.
We will be interested below only in the case when
$\vv \cap \Cal D = \emptyset $ i.e. when $\gb$
is not ``tightly attached'' to $\Cal D$.
\definition{$A_{\text{full}}(\gb)$ and other variants of $A(\gb)$}
In analogy to (3.21) define also the modified quantities
$A_{\loc}(\gb)$,\
$A_{\full}(\gb)$,
$A_{\full,\Cal D, \Lambda }(\gb)$\
as in the relation (3.21) but with the volume
$\vv$ in the second and the third sum on the right hand side of (3.21)
being replaced successively by volumes
$V(\gb)$,\ $\zv$,\
$\Lambda \setminus \supp\Cal D $. \
Corespondingly define, by (3.22), the quantities
$F_{\loc}(\gb)$,\
$F_{\full }(\gb)$,\
$F_{\full,\Cal D , \Lambda }(\gb)$
and also $F_0(\gb)$) (by taking $A_0(\gb)$ from (2.20)).
\enddefinition
\remark{Note} Assuming the existence of the cluster expansion for the
partition functions in the expression (2.21) for $F_{\text{formal}}$
one sees
that the true analogy of the quantity $F_{\text{formal}}$
is $F_{\full }(\gb)$, not $F(\gb)$. However,
$F_{\full }(\gb)$ is a {\it nonlocal\/} quantity
(though a very quickly converging sum of local quantities
$k_T$) and this would make the expansions
constructed below in Theorems 3, 4, 5 practically
useless. (However, we will return to the value
$F_{\full }(\gb)$ in later sections.)
On the other hand, $F_{\loc}(\gb)$ is a perfectly local
quantity, but sometimes it is ``too crude'' (and therefore
never used below) approximation
to $F_{\text{formal}}(\gb)$; such a
situation happens in the cases where there are {\it very\/} big
flat ``ceilings'' on the boundary of $V(\gb)$; a situation
having no analogy in the translation invariant
situation where the choice of $F_{\loc}(\gb)$ would be
without problems
because contours are ``crusted'' in that case.
The quantity $F(\gb)$
is a reasonable {\it compromise\/}
because it approximates \ $F_{\full }(\gb)$ \ with a great
accuracy ($\sim \exp(-\tau | \supp \gb|$) and at the same
time it is ``local'' in a reasonable sense. One could take
even smaller sets $\vv \supset V(\gb)$ to retain this accuracy --
but our
choice has an advantage to be
in conformity also with previous topological
constructions, namely
with our approach to the notion
of a contour.
The quantity $F_{\full,\Cal D , \Lambda }(\gb)$ is just
a temporary notation used in the
proof of Theorem 3 below.
\footnote{ One general ``philosophical'' remark: sometimes, one is fighting severe
technical problems in the \ps \
theory which however
are relevant only in volumes
which are really {\it astronomically large\/};
for example the problem
mentioned above (namely the ambiguity in the value of
different variants of $A(\gb)$)
is hardly of much relevance in volumes
of medium sizes like $10^{27}$ !}
\endremark
The forthcoming theorem
is an {\it essential\/} technical step in our procedure
of ``recoloring of an internal contour system''. This is further
developed by Theorem 4 and finally by Theorem 5 of the next section
where we return back to
horizontally invariant models.
\footnote{Thus, our Theorems 3 and 4 are just some
{\it technical steps\/} used merely
to prove Theorem 5. We admit that a more direct proof of Theorem 5
could be also given, by formulating Theorems 3 and 4 directly
for {\it collections\/} of compatible shifts of $\gb$,
thus
avoiding (at the price of more complicated formulation
of relations like (3.23), (3.36) etc.)
the use of horizontally noninvariant mixed models.}
\proclaim {Theorem 3}
Assume that we have a mixed model satisfying (3.1).
Let $\Lambda \subset \zv$. Let and $\gb$ and $\Cal D$
be two disjoint contour systems such that
$\Cal D \cup \gb$
is also an admissible contour system. Denote it
by $\Cal D\ \&\ \gb$, assume that
$\gb$ is ``strictly internal in $\Lambda$'',
satisfying the condition
$ \vv \cap \cap (\Cal D \cup \Lambda^c) =\emptyset$.
Let $\alpha \in \es $
\footnote { It is sufficient to have the condition
$\alpha \in \es$ only ``locally''
on $\partial (\Lambda)^c$, and actually we will need this
more general case at the very end of our paper, when
deriving the relation (3.93).}
be a boundary condition on
$\partial \Lambda ^c$ which is in conformity
with both $\Cal D$ and $\Cal D\ \&\ \gb$
i.e. such that $(x_{\Cal D\&\gb})^{\text{ext}} =
(x_{\Cal D})^{\text{ext}}=\alpha$. Then
$$
Z_{\Cal D\&\gb}^{\alpha }(\Lambda )=
Z_{\Cal D}^{\alpha }(\Lambda )
\exp(-F(\gb))
\exp(\sum k_T^{\text{corr}})
\tag 3.23
$$
where $k_T^{\text{corr}}$ (it is given below by
(3.26) and (3.27)) depends on
$x_{\Cal D \& \gb}|\vvt$
\footnote{For typhographical reasons,
$x_{\gb}|V$ denotes here (again) the restriction $y_V$
of the configuration $y = x_{\gb}$.}
and $x_{\Cal D}|\vvt$,
$\vvt = \cup_{G \in T} \vvg$, and the summation is
over all
$T\ssubset \Lambda, T\cap \cap \vv^c \ne \emptyset $
``touching $\gb$''
such that $\dist(T,\supp\gb) \leq 4 \diam T$.
The quantities
$k_T^{\text{corr}} $ satisfy for each $T$ the bound
$$ |k_T^{\text{corr}}| \leq 2\,\varepsilon ^{\conn T}. \tag 3.24$$
\endproclaim
\demo{Proof}
Write $\gamma \delta $ resp\. $\delta $ instead of
$x_{\Cal D\&\gb}$ resp\. $x_{\Cal D}$.
Write $Z_{\Cal D\&\gb}^{\alpha }(\Lambda )$ as
$$
\aligned
&\exp(-E(\gb)-E(\Cal D ))
\exp(-\sum_{t\in\Lambda }e_t(\gamma \delta ))
\exp(\sum\Sb T\ssubset\Lambda \setminus (\supp \Cal D
\cup \supp \gb)\endSb
k_T(\gamma \delta )
)=\\
&=\exp(-E(\Cal D ))
\exp(-\sum_{t\in\Lambda }e_t(\delta ))
\exp(-F_{\full,\Cal D , \Lambda }(\gb))
\exp(\sum\Sb T\ssubset\Lambda \setminus \supp \Cal D
\endSb
k_T(\delta )
)=\\
&=Z_{\Cal D}^{\alpha }(\Lambda )
\exp(-F(\gb))
\exp(
\sum
\Sb T\ssubset \Lambda \\
T\ccap ( \vv)^c \ne \emptyset \\
T \ccap (\supp\Cal D \cup \supp \gb) = \emptyset
\endSb
k_T(\gamma \delta )-
\sum
\Sb T\ssubset \Lambda \\
T\ccap( \vv)^c \ne \emptyset \\
T \ccap (\supp\Cal D) =\emptyset
\endSb
k_T(\delta )
)
\endaligned .
\tag 3.25
$$ and this relation
\footnote{In the last expression, we just write the detailed
expression of $F(\gb) - F_{\full,\Cal D , \Lambda }(\gb)$.}
proves (3.23) with the following choice of the
quantities $k_T^{\text{corr}}$ (recall that the ``old''
quantities $k_T$ satisfy (3.1), hence we have the bound (3.24))
$$ k_T^{\text{corr}}=k_T(\gamma \delta ) - k_T(\delta )
\tag 3.26$$
if $ T\ssubset \Lambda\, , \, \,
T\ccap ( \vv)^c \ne \emptyset\, , \, \,
T \ccap (\supp\Cal D \cup \supp\gb) = \emptyset,
\vvt \cap V(\Gammab) \neq \emptyset $, resp.
$$
k_T^{\text{corr}}=-k_T( \delta )
\tag 3.27 $$
if $
T\ssubset \Lambda\, , \, \,
T\ccap( \vv)^c \ne \emptyset \, , \, T \ccap \supp
\Cal D = \emptyset \, ,
\ T \ccap \supp\gb \neq \emptyset $.
The last condition $ \vvt \cap V(\Gammab) \neq \emptyset$
in (3.26) is obviously
necessary for a {\it nonzero\/} result in (3.26).
To check that such a condition implies also the
condition
$ \dist(T,\supp\gb) \leq 4 \diam T$ (the latter
condition we
automatically have in (3.27), with the constant 1
instead of 4)
combine the condition $\vvt \cup V(\gb) \ne \emptyset$
with the
condition $T\ccap ( \vv)^c \ne \emptyset$. Then use
the following observation. Consider what the relations
stated above
for $T$ mean for any element $G \in T$
(warning: $G \ne \supp \gb$):
Namely, then (the following argument is for a cluster
$T =\{G,\emptyset\}$, the estimate in the
general case is even better)
the validity of
$$ G\cap\cap (\vv)^c \ne \emptyset
\ \ \& \ \vvg \cap V(\gb) \neq
\emptyset \tag 3.28 $$
would imply (draw a picture) that the integers
$d,g,m$ denoting the distance of $G$ from
$\supp \gb$ resp. the diameter of $G$ resp.
the diameter of $\supp \gb$ would satisfy the
following relation:
$ 3g > m \ $ . Thus, $d
\leq 1/2 m + g
\leq 5/2 g < 4g$.
\footnote{Notice that any point of $V(\gb)$ has a distance
at most $1/2 \ m$ from $\supp \gb$ and hence $d < 1/2 \ m +g$.
Also, $\vv = V_{\updownarrow}(V(\gb)$)
and the distance of $G$ from $(\vv)^c$
is at most $ g$ if $G \cap \cap (\vv)^c \ne \emptyset $.}
\enddemo
\definition{Notation}
A subsystem $\gb$ of a contour system $\gb'$ satisfying
the condition of ``isolation'' $
\supp \gb \ccap (\supp \gb' \setminus \gb) = \emptyset$
will be called a {\it strictly interior\/} contour
(sub)system of $\gb'$.
\enddefinition
\proclaim{Theorem 4}
Assume that we
have a mixed model $\Cal M$ satisfying (3.1).
Consider the partition functions (3.19),
with additional summation over
$\gb$ (with a fixed $\supp \gb = G$):
$$
Z_{\Cal D, G}^{\alpha}(\Lambda) =
\sum_{\gb:\supp\gb=G}Z_{\Cal D \&\gb}^{\alpha }(\Lambda)
\ \ \ \ \text{and }\ \ \ \
Z_{\Cal D, [\,G\,]}^{\alpha}(\Lambda) = Z_{\Cal D}^{\alpha}(\Lambda)
+ Z_{\Cal D, G}^{\alpha}(\Lambda)
\tag 3.29$$
where $\Cal D \& \gb$ is a contour system
such that $\gb$ is
its strictly interior contour subsystem.
The partition functions above
correspond to the events ``\ $\Cal D \ \& \ \gb
$ appears'' resp. `` $\gb: \supp \gb = G$
could appear with $\Cal D$''.
These partition functions can be expressed correspondingly
as
$$
Z_{\Cal D, G }^{\alpha }(\Lambda )=
(\sum_{T \ssubset \Lambda:G\in T} k_T^+(\delta))Z_{\Cal D}^{\alpha}(\Lambda) ,
$$
$$ Z_{\Cal D, [G]}^{\alpha}(\Lambda)=\exp(\sum_{T
\ssubset \Lambda:G\in T} k_T^*(\delta))
Z_{\Cal D}^{\alpha}(\Lambda)
\tag 3.30
$$
respectively; recall that $G\in T$ means
that $G$ is the core of the cluster T
and $k_T^+(\delta)$ resp. $k_T^*(\delta)$ are some new cluster terms, described
in detail in the proof.
The leading new quantity $k_G^*(\delta) = k_G^+(\delta)$
is equal to
$$ k_G^+ = k_{G}^+ ( \delta) = \sum_{\gb:\supp \gb = G} \exp
(-F(\gb)).
\tag 3.31 $$
where the sum is over all contours
having a given support $G$ and a given
external colour $\delta$ determined by $\alpha$
and $\Cal D$.
The remaining new quantities $k_T^+$ obey the bounds,
for any new cluster of the type
$T =(G, \{T_i\})$
(where $T_i \ ;i = 1, \dots, m $ are clusters of
$\Cal M$)
$$ |k_T^+(\delta)| \leq k_G^+ (\delta)\ 2^m \varepsilon ^{\ \sum_{i=1}^m \conn T_i}\,
\tag 3.32 $$
and analogous bounds are valid for $k_T^* $.
\footnote{See (3.41) for a more precise estimate
on the right hand side; also for more
complicated clusters $T$.}
The quantities $ k_G^+,\ k_T^+,\ k_T^* \ $ do not already depend on $\gb$;
they depend only on $ G$ and
on the values $\delta_G$
\footnote{See Theorem 3; $k_G^+$ does not depend on the
interior colour of $\gb$
in contrast to $\exp(-F(\gb))$.}
and they are translation invariant in the sense of (3.16).
Moreover, if $\ k_G^+(\delta) $ satisfies
a bound of the type $$k_G^+ (\delta)\leq \varepsilon ^{\Conn G}
(\varepsilon')^{\diam G}\tag 3.33 $$
with another small $\varepsilon'$
(this is a simpler variant of (3.8))
then the validity of
(3.1) in the given mixed model $\Cal M$ implies
its validity
\footnote{ Even with some ``reserve'',
see (3.41).} also for the new quantities
$k_T^+$ and $k_T^*\ $.
\footnote{Theorem 5, next section, formulates
a ``horizontally invariant''
modification of Theorem 4.
The constant $\varepsilon$ is used throughout in
bounds like (3.1) and its value is assumed to be sufficiently
small, to guarantee Proposition 2, section 1 and therefore
the quick convergence of all infinite sums.}
\endproclaim
\definition{How to use Theorem 4}
The procedure called ``recoloring of a contour system''
described by formula (3.30) will be used later repeatedly many times
for all the ``smallest possible'' strictly interior contour systems
$\gb$.
The new cluster quantities $k_T^*$ will play the same role as the
``old'' quantities $k_T$ and therefore it is
crucial to ensure (through (3.33))
the validity of (3.1) for them.
This means that the new mixed model
constructed by the formula (3.30) (and which has a new, richer family of
cluster fields $\{k_T\} \& \{k_T^*\}$ but which does not yet allow
$\gb$ as a strictly interior contour system of its configuration)
is of the {\it same type\/} as before, satisfying again the estimate
(3.1).
However, first we will replace
(3.33) by a more comfortable estimate: \enddefinition
\definition{Recolorable contour systems}
This notion will be defined (in view of Theorem 5) only for
horizontally invariant mixed models. If we have a
possibly noninvariant
model (like here in Theorems 3, 4) we define
the {\it (horizontally) invariant part\/}
$\Cal M_{\text{inv}}$ of the model $\Cal M$ by
\footnote{We do not mean here, of course, the
noninvariancy caused by the fact that we are in
a finite $\Lambda$!}
discarding all the values $k_T(y)$
which are {\it not\/} horizontally invariant.
Now we will say that a strictly
interior contour system $\gb$ is {\it recolorable\/}
(more precisely $\hat \tau$--recolorable) in the mixed model
$\Cal M$ if the inequality ($F_{\text{inv}}$ is taken in
the model $\Cal M_{\text{inv}}!$)
$$ \sum^{x_{\gb}^{\text{ext}}= y}_{\gb :\supp \gb = G}\exp (-F_{\text{inv}}(\gb)) \leq \exp(-\hat
\tau \conn G)
\tag 3.34 $$
holds with a large $\hat \tau$, for each possible
external
colour $y = x_{\gb}^{\text{ext}}$ given on on $\vv$.
\enddefinition \remark{Notes}
1. Notice that this is the requirement on the
{\it set \/} $G$ and the external colour $y$,
not on a particular contour system $\gb$ -- though practically this is closely related
to
saying that for each $\gb$ with the same support $G$,
one has a bound, with some other large $\hat \tau^*$,
$$ F_{\text{inv}}(\gb) \geq \hat \tau^* \conn \gb . \tag 3.34*$$
2. One should always have in mind that the property ``to be
recolorable'' is rather sensitive (for contour systems of {\it very\/} large
size, of course) to the boundary
conditions. The mere knowledge of the external
colour of $\gb$ only ``at the
vertical level of
$V(\gb)$''
may be insufficient to
decide the recolorability. In general, the external
colour of the whole volume $\vv$ (some logarithmic
neighborhood of $
V(\gb)$ would suffice, in fact) must be known.
3. In practice, the requirement (3.33) can be checked
through the recolorability (3.34) of $\gb$,
with $\hat \tau $
satisfying (with some ``reserve'') the inequality
(compare (3.31) and (3.33))
$$ \exp(-\hat \tau\conn G) <
\varepsilon ^{\Conn G}
(\varepsilon')^{\diam G}. \tag 3.35
$$ Then
the relation (3.35)
implies also (3.33). Namely, only ``slightly
noninvariant''
models,
with $F(\gb) - F_{\text{inv}}(\gb)$ being
of the order $\varepsilon^{\diam \gb}$ only,
appear when applying Theorem 4 successively
(by recoloring all the shifts of some $\gb$,
see the proof of Theorem 5)
to some invariant mixed model.
4. Thus, the bound (3.35) relates the appropriate
choice of the constants
$\varepsilon, \hat \tau$\ in (3.1) and (3.34).
The latter condition
is a {\it Peierls type condition\/}
for the quantity $F(\gb)$.
The possible choice of $\hat \tau = c \tau $
(such that (3.34) could be checked;
$\tau$ is from (2.14))
will be discussed below (see (3.59); the convenient choice of the constant
being $c= 1 / 12\nu $).
\footnote{It is not at all necessary that $\hat \tau$
in (3.34) would be absolutely the same for
all $\gb$.
For
different shapes of $\gb$ one can take different
(large) $\hat \tau$. However,
it is technically important (for easy formulation
of Theorem 5) that recolorability would hold
for all the
horizontal translates of $\gb$ at once.}
\endremark
5.
One should also emphasize that when performing later
our successive
process of
``recoloring of all
(recolorable) $\gb$'', our procedure
will be organized in such a way (see the
forthcoming section) that quantities $k_T^*$ with
{\it new\/} clusters $T$ (nonexistent with
nonzero $k_T$ in the previous mixed model) will appear at each
stage of the construction.
\endremark
\demo{Proof of Theorem 4
} The proof
is based on Theorem 3
and Proposition 2, section 1:
By (3.23) we have the relation
$$
Z_{\Cal D, G }^{\alpha }(\Lambda)=
\sum_{\gb:\supp\gb=G} Z_{\Cal D \&\gb}^{\alpha }(\Lambda)=
Z_{\Cal D }^{\alpha }(\Lambda) \sum_{\gb}
\exp(-F(\gb))
\exp(\sum_T k_T^{\text{corr}})\, .$$
Writing $\exp(-F(\gb)) = \xi _{\gb} k_G^+$ and expanding the
exponential this can be written as
$$ Z_{\Cal D, G }^{\alpha }(\Lambda)
= Z_{\Cal D }^{\alpha }(\Lambda) \biggl(\sum_{\gb} \xi _{\gb}
\sum_{k=0}^{\infty} \sum\Sb (T_1, \dots, T_k)\\
(n_1, \dots, n_k) \endSb
\prod_{i=1}^{k}\frac{1}{n_i!}(k_{T_i}^{\text{corr}})^{n_i})\biggr)
k_G^+ =
Z_{\Cal D }^{\alpha }( \Lambda) \ \sum_T k_T^+ .\tag 3.36$$
The summation is over new clusters T (see below)
satisfying analogous properties as in
(3.23) and the new values of $k_T$
are defined here as
\footnote{Notice that whereas
$F(\gb)$ depends on the values $x_{\gb}| \vv$ ,
in particular on the
``interior colour of $\gb$'', the quantity $k^+_T$
depends only on $G$
and the ``external colour'' of $\gb$ restricted to
$\vv$.}
$$
k_T^+ = \sum\Sb \gb:\supp \gb = G\endSb
\xi _{\gb} k^+_G \prod_{i=1}^{k}\sum_{n_i \geq 1}
\frac{1}{n_i!}(k_{T_i}^{\text{corr}})^{n_i}
. \tag 3.37$$
The clusters $T$ are defined (for $n_i=1$; notice that there is a summation over
$n_i \geq 1$) as
$$ T= (G, \{T_i\}) . \tag 3.38 $$ the most
important term
\footnote{ This is the point where our hierarchical construction
of clusters is rather important, to assure that we really have
{\it new\/}
clusters here, which were not used before (e.g. in (3.23)).
Not having this, the condition
(3.1) could have been spoiled after adding several new values
of $k_T$ for the same cluster $T$ !}
corresponding to
the empty collection of $\{T_i\}$.
Notice that the expression of $k_T^+$ by values $k_T^{\text{corr}}$
is not exactly as (3.7), Proposition 1 of Section 1.
However, it is straightforward
to adapt the corresponding estimates noticing that $\sum_{\gb} \xi _{\gb}
= 1$. Thus, one obtains (3.32) (the term $2^m$
is from (3.24) !) and then, from (3.34) and (3.35),
also (3.1) for the
quantity $k_T^+$.
Namely the term (emerging,
for the new cluster $T = \{G,\{T_i\}\}$,
if we directly substitute
(3.32) and the bound (3.33) for $k_G^+$ into (3.37))
$$ 2^n \ \varepsilon^{ \sum_{i=1}^n \conn T_i + \conn G }
\ (\varepsilon')^{\diam G} $$
is surely smaller than $\varepsilon^{\conn T}$
as some inspection
of the notion of $\conn T_i$
shows.
The expansion (3.31) is now obtained by taking
logarithms. We get \ \ $\log Z_{\Cal D, [G]}^{\alpha}(\Lambda) = $
$$ \log Z_{\Cal D}^{\alpha}(\Lambda)+
\log(1+ \sum_{(T_1,T_2,\dots,T_n) \ssubset \Lambda}^{ (T_1,T_2,\dots,T_n)\owns G} k_{(T_1,T_2,\dots,T_n)}^+) =
\log Z_{\Cal D }^{\alpha }(\Lambda)+ \sum_{(T_1,T_2,\dots,T_n) \ssubset
\Lambda}^{(T_1,T_2,\dots,T_n)\owns G}
k_{(T_1,T_2,\dots,T_n)}^{(*)}
\tag 3.39 $$
where
\footnote{ For typhographic reasons,
we write part of the requirements on the summands
above the sum.}
(notice that {\it all\/} the clusters $T_i$ mentioned in these
relations
contain the set $G$)
$$
k_{(T_1,T_2,\dots,T_n)}^{(*)}= \frac{(-1)^{n-1}} {n}
\prod_{i=1}^{n}k_{T_i}^+
\tag 3.40$$
for any ``cluster'' $(T_1, T_2, \dots, T_n) $;
and one has to modify
correspondingly this formula if multiple copies of one cluster
$T_i$ appear in $T$.
However, we will {\it not\/} define
``clusters'' of a type $(T_1,T_2,\dots,T_n)$. Instead, we perform
a {\it partial summation\/} over
all these collections $(T_1, T_2, \dots, T_n) $
with the same $ T =\cup \ T_i$ in the relation
\footnote{By $\cup \ T_i$ we mean the cluster $(G,\{T^j_i\})$
where $T_i =(G,\{T^j_i\})$; each $T^j_i$
being counted only once in $T$.}
(3.39),
namely we put
$$ k_T^* = \sum_{ (T_1,T_2,\dots,T_n)} k_{(T_1,T_2,\dots,T_n)}^{(*)} \tag 3.40' $$
where the summation is over all
$(T_1, T_2, \dots, T_n) $ with
the same union $T =\cup \ T_i$ .
Now, for any such $n$ -- tuple
$(T_1, T_2, \dots, T_n)$ (recall that $G\in T_i$ for
each $ i$)
one obtains, after some inspection, the bound
$$ |k_{(T_1,T_2,\dots,T_n)}^{(*)}| \leq (k_G^+)^n\
\varepsilon ^{\sum_{i=1}^n \conn ^+ (T_i)}
\ \ \ \ \ \text{i.e.} \ \ \ \ \ k_T^*\leq \varepsilon ^{\conn T}\tag
3.41$$
where $\conn^+ ( T) $ denotes the quantity $ \sum_j \conn T_j$ for
$T=(G, \{T_j\})$; the last inequality follows from (3.34)
and (3.35) by a similar argument as we used in the estimate of the
values (3.37) above.
This
concludes the proof of Theorem 4.
\enddemo
\vskip1mm
\head 4. Recoloring: towards a new stratified mixed model
\endhead
The aim of this section is to formulate a procedure (based on
Theorem 4)
which converts a given {\it horizontally translation invariant\/}
mixed model into a new horizontally
translation invariant model, having the same ``diluted ''partition functions
but with a {\it smaller\/}
set of allowed configurations (and with a {\it richer\/} set of clusters $T$
having
nonzero contributions $k_T$; the ``old'' nonzero values $k_T$ being
kept at the {\it same value\/} as before).
Such a transformation of the model could be characterized
as the ``removal, from the model,
of all configurations which have a shift of $\gb$
among its strictly interior contour systems''.
Recall the ordering $\prec$ (and
$\prec \prec$) of contour systems we
introduced in section III,1.
We will say that a recolorable contour system $\gb$
is {\it smallest
recolorable\/} contour system of the given mixed model
if it is strictly interior in the considered volume
$\Lambda$, recolorable and moreover there is no smaller,
in the ordering $\prec$ (resp. $\prec \prec$
if we work with horizontally invariant models)
, strictly interior
recolorable
$\gb'$ which could appear in
some configuration of the given mixed model.
\remark {Note}
Below we will use the recoloring step
formulated by Theorem 5 {\it successively\/},
according to the growing ``size'' of the smallest recolorable
contour systems $\gb$ which have to be
recolored.
Moreover, the mixed models studied by us later will appear as the
result of successive recolorings applied to some given
\ps \
abstract model; the configuration spaces of the mixed models thus
obtained will be defined in terms of
requirements on the {\it size\/}
of the smallest interior recolorable
contour systems
of the configuration. See the forthcoming sections for details.
\endremark
\definition{The notion of an equivalent mixed model}
Recall that we say that a cluster $T$ {\it contains\/} a contour system
\footnote{Of course this is a slight abuse of notations
because clusters are made of {\it sets\/}
$\supp \gb$, not of contours $\gb$. However,
we always keep in mind the
{\it external colour\/} $y$
of $\gb$.}
$\gb$ if the set $\supp \gb$
was used, with the external
colour $x_{\gb}^{\text{ext}}$, as a core of some
intermediate cluster in
the recursive construction of $T$.
Two mixed models will be said to be equivalent if
all their strictly diluted partition functions (for all
finite volumes and
boundary conditions $y \in \es$) are the
{\it same\/}.
We notice that usually, the configuration space of one of these two mixed models
will be a suitable {\it subset\/} of the configuration space
of the
other model; on the contrary the collection of clusters
with nonzero $k_T$ for the model with
a smaller configuration space will be usually {\it richer\/},
including also
some clusters
containing contours which are present only in the model with
a bigger configurations space. This is the case of
the following result.\enddefinition
\proclaim{Theorem 5}
Assume that we have a horizontally translation invariant
mixed model satisfying the condition (3.1). Consider any
finite volume $\Lambda$. \footnote{Theorem 5
is applied in all
finite volumes {\it at once\/}, for any fixed
$\supp \gb$ and $y_{\vv}$.}
Let $\gb$ be a recolorable \footnote{ In particular,
we may assume that $\supp \gb$ is smallest possible in the relation
$\prec$; \
this will be the case needed in the
applications of Theorem 5 below. We assume that $\hat \tau$
from (3.34) is sufficiently large !} contour system
whose horizontal shift can appear as a strictly interior
contour system
of some configuration of the model. Denote by
$y$ its external colour.
Assume that $k_T(y) = 0 $ holds for all clusters $T$
containing a shift of
$\supp \gb$.
Then there is an equivalent mixed
model having the following properties :
\roster
\item Its configuration space is the collection of all configurations
of the original mixed model which do not contain a
horizontal shift of some $\gb'$, $\supp \gb =\supp \gb'$
with the same external colour $y$,
among its possible strictly interior recolorable
subsystems.
\item If $T$ is a cluster not containing a horizontal
shift of
$\supp \gb$
then the value of $k_T(y)$ in the new mixed model is the same as before.
\item If $T$ contains a horizontal shift
of $\supp \gb$ then the new value of $k_T$
satisfies the condition (3.1), too, assuming that $\varepsilon$ and
$\hat \tau$ are such that (3.34) holds.
\endroster
\endproclaim
\remark{Note}
Notice that when applying Theorem 5
to one particular $\gb$ it does not mean that some
other $\gb'$ with the same support
$\supp \gb' =\supp \gb$ with
the external colour $y'$ differing from
$y$ on $\vv$ would not still {\it remain\/} in the resulting
mixed model. So we have to use Theorem 5
{\it repeatedly\/} again -- to remove
all the recolorable systems $\gb$ with the same support
$\supp \gb$ but with different external colours $y$.
The recoloring of various contour systems $\gb$
with different externals colour $y$ and the same $\supp \gb$
is made independently, the corresponding
expansions not affecting
each other!
\endremark \demo{Proof} This follows
from Theorem 4 if we use it successively in the following
way: take the (completely ordered in $\prec$ ) sequence of
all shifts of $\gb$. Given $t \in \zv$ consider an
intermediate
``$t$--th model'' \footnote{This has sense in any
{\it finite\/} volume $\Lambda$,
and the cluster terms $k_T^*$ do {\it not\/}
depend on the volume; however the summation
over $T$ depends on $\Lambda$.}
defined below
which has the configuration space defined by the requirement
that exactly those configurations of the original mixed model are allowed
for which {\it no\/} strictly interior contour system
$ \gb + t'$ such that $ t' \prec_l t $
exist.
If $s$ is the nearest greater point
to $t$ ($ t \prec s $) in
the given finite volume $\Lambda$
then we define the transition to the $s$--th model by the very
procedure
described by Theorem 4.
It is straightforward to check the translation
invariance (3.16) and uniqueness of the definition (not depending on
the actual volume $\Lambda$; recall that $T \ssubset
\Lambda$) of all the new
quantities $k_T$ thus obtained.
\footnote{ There are modifications possible
in the method we used to prove Theorem 5.
Namely, it is possible (and it is, in fact, apparently a more
standard way how to deal with these cluster expansions) to reformulate the
Theorem 3 and Theorem 4
above into a single statement describing
{\it simultaneous recoloring\/} of all the shifts of $\gb$
{\it at once\/}. We do not follow such a (more direct, but with slightly
more complicated formulas) approach in this paper.
However, such an approach is used also in the recent lectures \cite{ZRO}
(in a simplest possible form, we believe)
and in future publications, we plan to replace the arguments based on the successsive
use of the lexicographic order by this more standard approach.}
\enddemo
\head 5. Small and extremally small contour systems
\endhead
We are still working with a {\it general mixed
model\/}. Only later we will explain the relevant choice
of a
mixed model in a concrete situation; this choice
will be always given as a suitable
partial expansion of the original abstract
Pirogov -- Sinai model.
The successive application of the recoloring procedure
constructed in the preceding chapters (culminating
in Theorem 5) will finally lead, in any volume,
to a family (indexed by elements of $\es$)
of mixed models where {\it no\/} recolorable systems will
be left! The reader is advised to look briefly to the section 8
and to look at the formulation and the proof of Main Theorem
-- to see the important consequences of this fact.
In the meantime, in sections 5 to 7, we will investigate
the notion of a recolorable contour system (more precisely
the related
notion of a small resp. extremally small contour system introduced
below) in a more depth. We will construct
an important concept of a
``skeleton'' of a contour system $\gb$;
this notion will be investigated using
some useful supplementary
``topological'' results derived later,
namely Theorems 6 and 7.
All this will be needed in the proof
of the Main Theorem. Possibly, our discussion of the notions of a ``small'' and
of an ``extremally small'' system, and our definition
of the notion of a ``skeleton'' of $\gb$
(notice that we will introduce
there another testing quantity $A^*(\square)$, as a more
careful alternative to $A(\gb) $) is
slightly more detailed than it is absolutely necessary --
if a shortest possible proof of Main Theorem
is required.
However, we are keeping all these seemingly unnecessary
details
here -- also because we expect our more
detailed exposition will be useful not only here
(giving more information
and estimates with better constants
compared to a more crude method) but also
in the future
investigations of the ``metastability'' problem and
in the study of the
completeness of the phase picture constructed by
our Main Theorem.
The reader interested in acquiring the {\it
idea of the proof\/}
of Main Theorem can go now directly to Section 8,
omitting even the
very notion of a small contour system but finally realizing that
some variant of such a notion
(and of Theorem 7) is needed there!
\definition{ Preparation: Relations between different
variants of $A(\gb)$}
\enddefinition
Recall the definition of $A(\gb)$, $A_{\text{full}}(\gb)$,
and of
the corresponding quantities $F(\gb)$
from Section 3.
When checking the condition (3.34) or (3.34*)
one needs (at least for contour systems whose energy is not much
above the bound assumed in (2.14)) an
inequality
$$ A(\gb) \leq \tilde \tau |\supp \gb | $$
where $\tilde \tau < \tau$ is a suitable
constant ``not too close to $\tau$'' ($\tau$ from (2.14)).
\footnote{In fact, there is quite a freedom in the choice of all these
constants $\tilde \tau,\ \hat \tau, \hat \tau^*$
from (3.34,34*))
resp. from the relation (3.49) below and
the difference between various choices
of $A(\gb)$ (and, accordingly of $F(\gb)$)
which was so
important in Section 3 will be quite irrelevant here,
except of the choice of
$A_{\text {loc}}(\gb)$ which would be too rough in
situations
where some {\it really big\/} ``ceilings'' of $\gb$ appear.}
In these estimates, it will be more convenient
to work with quantities of the type $A_{\text{full}}$
instead of $A$. Fortunately, both quantities
are practically the same for any $\gb$. More precisely we
have the
following bound which is easily obtained if
we notice that the cluster terms $k_T$ acting in the
difference below -- compare (3.21) -- decay
exponentially quickly
with the distance of $T$ and $\supp \gb$
(for such an observation it is important to recall that
the set $\vv$ -- and not $V(\gb)$, for example--
was used in the definition of $A(\gb)$!).
\proclaim{Proposition 1} For any contour
system $\gb$ (and for any extension of the
boundary condition $x_{\gb}^{\text{ext}}$ to the whole
$\zv$), the quantities
$A(\gb)$ and $ A_{\text{full}}(\gb)$ satisfy the bound
$$ | A(\gb) - A_{\text{full}}(\gb)| \leq (\varepsilon')^{\diam \gb}
|\supp \gb| \tag 3.42
$$
where $\varepsilon' = C \varepsilon$ and $\varepsilon$
is from (3.1).
\endproclaim
The quantities $ A_{\text {full}}(\gb) $ (and, even more
importantly, the modified
quantities $\Am(\square)$ introduced below) will
be used instead of $A(\gb)$ in these final sections.
All the
bounds of the type $ A_{\text{full}}(\gb) \leq \tilde \tau |\supp \gb | $
will be studied rather for them instead of
the original quantities $A(\gb)$.
Then we supplement these bounds by (3.42).
\footnote{Once again: the quantities $A(\gb)$ remain,
of course, to be used
in (3.22) but their test ``whether
they are dangerously big'' will be
done through the closely related but ``more
nicely looking''
quantities $A_{\text {full}}(\gb)$. }
The meaning of the quantities $A(\Gammab)$ is that they
give some information about the
``volume gain of the free energy''
caused by the fact that inside $\Gammab$, possibly some
``more stable'' regime is established. One could ask this question
in a more precise way: whether the regime which resides inside $\Gammab$
is the ``best'' possible one
and also what is the
``energetically optimal realization''
of such a contour system.
Fortunately, one does not need to investigate
these questions in more detail, in particular the question
``what is the optimal shape of a contour''
has not to be approached here.
On the other hand, the question ``what is
the best possible stratified regime to be found inside $\gb$'' {\it will\/} be
important in the investigations below and we will approach it
as follows:
We rewrite, from now on, the quantity $A_{\text{full}}(\gb)$ in a
more concise way,
replacing the sum over {\it cluster\/} quantities $k_T$ by
a more nicely looking (and more flexible) sum of suitable {\it point\/}
quantities.
These latter quantities are however nonlocal (but very quickly
converging limits of local quantities).
Introduce the following notations.
\definition{Free energy densities of the mixed model}
For any mixed model and any stratified configuration $y$ define the quantity
$$ f_t(y) =e_t(y)- \sum_{T: t \in T} \frac{k_T(y)}{|T|}. \tag 3.43 $$
For an arbitrary nonstratified
$x$, define $f_t(x) =f_t(x_t^{\text{hor}})$ where
$x_t^{\text{hor}}$ (we denoted it also as $\hat x_t$)
is the
horizontally invariant extension of the configuration
$x_{\{t_1,\dots,t_{\nu -1}
,(.)\}}$.
\enddefinition
\remark{Note}
These quantities will be very important in the sequel. However, in spite
of their ``physical'' meaning which we discover below
(they will be interpreted as the ``density, at $t$, of the free energy of the
metastable state constructed around $y$'') there is still some
arbitrariness in their definition:
For example, the modified quantity
$$\tilde f_t(y) = e_t(y) - \sum_{t\in T} k_T(y) \tag 3.44 $$
where the sum is over all clusters $T$ such that $t$ is the first
point of $\supp T$ in the lexicographic order could be used in the
same way. \footnote{ However,
the physically important quantities like
$\sum_t (f_t(z) -f_t(y))$ are the {\it same\/} for both alternatives,
whenever $y$ and $z$ are stratified and differing only
on some layer of a finite width.}
\endremark
\remark{Agreement}
Here and below we need to work with configurations
$y \in \es$ defined on the whole lattice $\zv$.
Let us make an agreement that whenever we have a stratified
configuration defined, at the moment, only in a partial way
(typical situation: the external colour of some
configuration defined in some finite volume)
then we extend it \footnote{ In fact, this is a
comparable act of arbitrariness like
that we used
in our
choice of the sets
$\vv$.}
in some way
to a configuration on the whole $\zv$.
The details of the
extension will be usually irrelevant.
\endremark
\proclaim{Proposition 2} For any contour system
$\gb$ and any extension of its
external colour $x_{\gb}^{\text{ext}}$, the quantity
$A_{\text{full}}(\gb)$ can be
expressed by the following formula
where $\varepsilon''= C \varepsilon^n$, $n$ being the
cardinality of a smallest possible cluster
appearing in (3.21)
\footnote{ The quantity $\varepsilon $ is from (3.1)
and
$C =C(\nu)$ is a suitable constant.}:
$$ A_{\text{full}}(\gb) =
A^*_{\text{full}}(\gb)
+\Delta(\gb) \ \ \ \ \ \ \text{where} \tag 3.45 $$ $$
A^*_{\text{full}}(\gb) =\sum_{t \in \zv} (f_t(x_{\gb}^{\text{ext}}) -
f_t(x_{\gb})) \ \ \ \ \ \ \text{and} \ \ \ \ \ \
|\Delta (\gb)| \leq \varepsilon'' |\supp \gb| . $$
\endproclaim
\demo{Proof}
This follows
from
the observation that any $k_T(x_{\gb})$, analogously
$k_T(y), y = x_{\gb}^{\text{ext}}$, $T \ssubset (\supp \gb)^c$ which
is counted in
$A_{\text{full}}(\gb)$ is counted also (exactly once!)
in $$\sum_{t \in \zv} (f_t(x_{\gb}^{\text{ext}})- f_t(x_{\gb})) =
\sum_{t \in \zv} (e_t(x_{\gb}^{\text{ext}})- e_t(x_{\gb})) +
\sum_t \sum_{T\owns t} \frac
{(k_T(x_{\gb}^{\text{ext}})^{\text{hor}} -
k_T(x_{\gb}))}{|T|}.$$
The correction terms
$k_T |T|^{-1}$) contributing to $\Delta$ thus arise
only from
clusters satisfying
the relation
$T\cap \cap \supp \gb \ne \emptyset$ and
when summed they give the bound (3.45).
Namely $A^*(\gb) - A(\gb)$ can be written, when inserting
(3.43) into it and rearranging the
terms, as a suitable
sum of $k_T(y)/|T|$ over clusters
$T$ whose distance to $\gb$ is comparable
to their diameter. \footnote{
This observation is immediate
for the
``clusters acting already in the sum for
$A_{\text{full}}(\gb)$''.
The remaining clusters (appearing only in the sums
for $f_t$)
must satisfy the conditions
$T \cap \cap \gb \ne \emptyset
$ and also
$\vvt \cap V(\gb) \ne \emptyset$
-- to give a nonzero contribution
$k_T(\hat x) - k_T(\hat x^{\text{ext}})$.
We noted in an analogous situation
of Theorem 3 that such clusters $T$ really have
a diameter comparable to their
distance to $\gb$.} The contribution
of these terms is like
$C \varepsilon |\supp \gb|$.
\enddemo
The forthcoming notion will be useful for understanding
what would be the ``best possible gain of free energy''
inside a given contour $\gb$ :
\definition{Configurations minimizing $f_t(x)$}
Given a configuration $x$ which is stratified outside of some
finite volume $V$
denote by
$x_V^{\text{best}}$
the configuration minimizing, at each vertical section
$(t_1,\dots,t_{\nu -1},(.))$,
the value
$\sum_{t_{\nu}\in \zet} f_t(x')$
under the condition that $x' =x$ outside of $V$.
\enddefinition
\remark{Note} This notion is close (but not identical)
to that of $x^{\text{best}}_{\gb}$ used above.
Namely one can consider a configuration
$x^{\text{best}}_V$ in a volume $V =V(\gb)$,
where $\gb$ is a contour system. However, we do not
fix neither the values of $\partial_r \gb$
nor $x_{V(\gb ) \setminus \supp \gb}$ here.
We will use the notation $x^{\text{best}}_V$
mostly for {\it cubes\/} $V$.
Notice also that in comparison to the formulation of the
Peierls condition in Part II, we
use
the quantity $f_t(x')$ instead of $e_t(x')$
in the definition of $x_V^{\text{best}}$. However,
$f_t$ is roughly equal to $e_t$ and the sum of the terms $(f_t -e_t)$
over $\supp \gb$ is again (like in (3.42))
of the order $\varepsilon' |\supp \gb| $. If
a contour $\gb$ is such that $\supp \gb = V$
and the outside colour of $\gb$ is
everywhere the same stratified configuration
$y \in \es$ then
the difference between our old value
of $x^{\text{best}}_{\gb}$ and the new one
$x^{\text{best}}_V$ is a
quantity really negligible when checking the validity
of Peierls condition.
\endremark
Using the quantities $f_t$, we now
want to define an alternative
(with the same intuitive meaning)
to $A(\gb)$ which will
be more flexible in the fortcoming estimates.
\definition{The quantity
$A^*(V,y)$}
Given any finite volume \footnote{We will use later this quantity not only
for volumes $V= \vv$ but also (and mainly) for
{\it cubes\/} $V$.} $V$ and any $y \in \es$ \
introduce the quantity
$A^*(V,y)$, more precisely $A^*_{\text{max}}(V,y)$ :
$$ A^*(V,y) \equiv A^*_{\text{max}}(V,y)
= \max\{ \sum_{t \in \zv} (f_t(y) -f_t(z))\}
=\sum_{t \in \zv} (f_t(y)-f_t(y_{V}^{\text{best}}))
\tag 3.47$$
where the maximum is taken over all $z$
which are equal to $y$ outside of $V $.
\enddefinition
Compare now (3.47) with (3.45), for $V\supset \vv$:
By Proposition 1 and 2 we have
$$
A(\gb) - A^*_{\text{max}}(V,y) \leq
A(\gb) -A^*(\gb) \leq
\varepsilon''' |\supp \gb| \tag 3.48$$
where $\varepsilon''' = C \varepsilon$
and $y$ is (any extension of)
the external colour of $\gb$.
\remark{Notes}
1. Our preference of this notion (to the previous
alternative
$A_{\text{full}}(\gb)$
which was defined with the help of summation over
{\it clusters\/} $k_T$
is mainly an
aesthetical one. Namely, the sums of {\it point\/} quantities will
be more convenient in later estimates.
Notice that now we
do {\it not\/} require any stratification of $z$ (and
of $y_{V}^{\text{best}}$) inside $V$.
Later (see the section Skeleton)
we will introduce, for technical reasons, some new, artificial
``contours'' of the model. These new contours will have
the shape of a (large) cube $\square$ living inside of some
stratified regime of the actual ``physical'' configuration;
their ``energy'' will be
defined (it will be of the order $\diam \square$ only)
just to compensate the ``volume gain inside $\square$\,''
(arising from
the fact that the configuration inside of such an artificial contour will
be assumed to ``jump freely into some better regime
inside $\square$\,''.
The interplay between these formal notions and a comparison
to the behaviour of
the real contours of the model
can be best studied
in the language of the quantities $\Af$ .
\newline 2.
Recall that the quantity $\Af$
requires the knowledge of $y$ in the whole $\zv$.
Thus, there is some small arbitrariness in the definition
of $\Af(V,y)$ because our $y$ is usually given only in some finite volume.
This arbitrariness is
compensated by the more transparent form
of the right hand side of (3.47) compared to (3.21).
\endremark
Thus, when estimating the size
of the quantities $A(\gb)$ we will work, from now on,
with the more convenient quantities $A_{\text{full}}(\gb)$
or even $\Am(V(\gb))$.
In fact, it turns out that it is reasonable
\footnote{They are some technical subtleties in this recommendation.
They will be more clear later, after defining the notion
of an extremally small system, in the proof of Theorem 6. See (3.55).}
to restrict the whole
discussion of the ``dangerously big values of $A(\gb)$''
to the {\it superordinated cubes only\/}:
\endremark
\definition { Small cubes of $y \in \es$}
1) A cube $\square$ will be called {\it small\/},
more precisely $\tau'$ -- small, with respect to
a configuration $y \in \es$ if the following inequality
\footnote{Do not care about the particular choice of the
constant $\tau' \approx \tau$ here.
{\it Any\/} sufficiently big constant would do the job.
On the other hand, the advisibility of our very choice of $\diam \gb$
will be clear only later. We mention that the choice of $\partial \square$
instead of $\diam \gb$ here
(such an alternative could maybe look more natural as the
quantities $|\supp \gb|$ appear otherwise everywhere whenever
the
energy of a contour system
is considered)
would cause technical difficulties later.}
holds :
$$ \Am(\square,y) \leq \tau'\diam \square \tag 3.49 $$
where $\tau'$ is suitably
(slightly) smaller than $\tau$, see below.
2)
In finite volumes $\Lambda$,
there is some ambiguity in the definition above
-- depending on how big a neighborhood $\Lambda \setminus \square$
of
$\square$ is given with a uniqely given value of
$y \in \es$ . (Obviously, a distance of a size
$\dist(\square,\Lambda^c) \approx \log \diam \square$ is
sufficient to give
a precision $\ll \diam \square$ in the value of
$\Am(\square,y)$ above.)
In the following we will call
more precisely $\square$ a small
cube of a configuration $y_{\Lambda}$, $y \in \es$
if (3.49) holds for
{\it any\/}
\footnote{ The usage of this notion will be
practically the same as if we would put
``for {\it some\/} extension of $y_{\Lambda}$''. We will work both
with the
implications of (3.49) as well as with the impplications
of its negation, so there is no logical
preference between ``some'' or ``all''
in the definition above.
Fortunately, this is not of much importance because
if we take a sufficiently (slightly !) smaller
cube $\square' \subset \square$ then the value of
$A^*(\square', y)$ is practically independent on the
extension of $y$ outside of $\square$,
as we already noted above.}
stratified extension $y$ of $y_{\Lambda}$ to
$(\square)^c$. \enddefinition
\definition {The choice of $\tau'$}
In the following we {\it fix\/} a suitable
(slightly smaller than $\tau$)
$\tau' < \tau$ guaranteeing the bound
$ \tau' < \tau - \varepsilon' - \varepsilon''
-\varepsilon'''$; compare (3.42), (3.45), and (3.48).
\footnote{Notice that while in the definition
of smallness we fix the choice of a constant
$\tau' \doteq \tau$, much smaller constant
$\hat \tau \doteq \tau/12 \nu$ has to be used, for technical
reasons, in
the definition of recolorability.}
\enddefinition
Say that a rectangle $\square$ in $\zv$ is the
{\it covering
rectangle\/} of a volume
$S
\subset \zv $
if \ $ \square$ is the smallest possible rectangle
(smallest in the
usual partial ordering $\prec$ determined
by the diameters of the considered sets)
which is a superset of $S$.
The {\it cube\/} having the same center as the covering
rectangle and also the same diameter
(recall that we have the $l_{\infty}$ norm everywhere,
so this cube contains the above rectangle as its subset)
will be called the {\it covering cube\/}
of $S$.
We will denote the covering {\it cube\/}
(rectangles would be inconvenient to work with, and therefore
they are abandoned below)
$\square$ by a symbol
$ \square(S)$.
Instead of $\square(\vv)$ we will also use
the notation $ \square(\gb)$.
\definition {Small contour systems}
If $\square$ is the covering cube of $\gb$ and
$y$ is the external colour of $\gb$
then we will say that
$\gb$ is {\it small\/} if $\square$ is small with respect to
$y$.
More precisely
we will restrict this notion only to {\it strictly interior contour subsystems\/}
$\gb$
of a given admissible system $\Cal D$ in $\Lambda$, and we will say that
$\gb$ is {\it small in\/} $\Lambda$ if
the covering cube $\square(\gb)$ is small for
each stratified extension of
the exterior colour of $\gb$ induced by $\Cal D$ and $y$.
\enddefinition
\remark{Note}
The property ``to be small''
will {\it not\/} be formulated for
contour systems $\gb \sqcap \Lambda^c \ne \emptyset$.
It is easy to see
(it is just the monotonicity of $A^*$
with respect to a growing volume)
that for any small $\gb$
we have the inequality, with $y$ denoting the external colour
of $\gb$ (extended to the whole $\zv$, otherwise
some small corrections have to be added)
$$ A^*_{\text{max}}(V(\gb),y) \leq A^*_{\text
{max}}(\vv,y)
\leq A^*_{\text{max}}(\square(\gb),y)
\leq \tau' \diam \gb.
\tag 3.50 $$
Complement this with (3.42) and (3.45)!
The idea now is, roughly speaking, that all
small contour systems
should be
{\it recolorable\/}. This is obviously true
for contour systems consisting of {\it one contour only\/}
because then
we have from (3.50) and (3.42)
the following inequality (see (2.15)):
$$ F(\gb) = E(\gb) - A(\gb) \geq (\tau^*
-\varepsilon)|\supp \gb| -
\tau' \diam \gb
\geq { \tau^* \over 2} |\supp \gb| \tag 3.51 $$
and the last term is bigger than, say
$ \tau^* /12\nu \ \conn \gb$ because
of Theorem 7 below.
However, we will often need to ``recolor'' also some {\it more
complicated\/} strictly interior contour systems i.e. systems of precontours
$\gb$ with unclear
relation between
$|\supp \gb|$ and $\conn \gb$. In such a case,
the corresponding more general argument valid for any
contour system
i.e. for any
admissible system of precontours $\gb$ will be developed in (3.54) below. However,
the notion of ``smallness'' has to be sharpened there
and the arguments (3.51) should be replaced by some
more detailed bounds
given below, in (3.60).
\definition{ Extremally small contour systems}
We will say that a small, strictly interior
contour system $\gb$ of a configuration
$(x, \Cal D)$ is {\it extremally small\/}
(more precisely $\tau'$--extremally small)
in $\Cal D$
if it is small and moreover if the following requirement for
$\gb$ is satisfied: for {\it no\/} strictly
interior contour system $\gb' \subsetneqq \gb$,
$\gb'$ is extremally small.
\footnote{One could substitute here
``small'' or even ``recolorable'' $\gb'$
(instead of extremally small $\gb'$)
without changing the sense of the definition.
However, a modification of such a notion
called
``extremal recolorability'' would be quite impractical.
We {\it will need\/} to know that extremal
smallness implies
recolorability!}
\enddefinition
\remark{Notes} {\bf 0.} A contour system $\gb$ will be later
called {\it removable\/}
if it is either extremally small or (more generally, see
below) recolorable
and it is moreover the smallest
one, in the ordering $\prec \prec $
among such contour
subsystems of a given mixed model.
We may inform the reader in advance
that below we prove that the
extremal smallness of $\gb$ implies its {\it recolorability\/},
and therefore also its removability
in the case of a smallest
possible size of $\gb$! However, smallness
is a much nicer property to work with; to check it suffices to look
only on the covering cube $\square(\gb)$ of $\gb$ and to
estimate the quantity $A^*(\square)$,
which is a more pleasant task than to estimate
$F(\gb)$.
The {\it minimality of size\/} in the definition above just reflects
our desire to organize the recoloring
by suitable {\it induction over the size of $\gb$\/}.
\newline {\bf 1.} We can claim now
that any contour system $\Cal D$ has the following property:
after the removal of a removable contour system $\gb$
(more precisely, of all the shifts
$\gb' \sqsubset \Lambda$ of $\gb$ according to Theorem 5)
from $\Cal D$, no
removable
contour system $\gb'$ of $\Cal D$
remains such that $\gb' \prec \gb$, and the same
remains true if subsequent removals are applied.
This property will be rather convenient
\footnote{ It will cause all the newly defined
clusters to be the {\it standard\/} ones.}
later when applying successively the procedure
of recoloring, by Theorem 5, to a given model.
\newline {\bf 2.}
The adjective ``small'' resp.``extremally small''
has only a loose
relation to the actual
{\it size} of $\supp\gb$. It is in accordance
with the usage of this term (and also of the related, perhaps
confusingly sounding term ``stable'') in [Z].
There are other adjectives used to describe such a property
in the literature -- like ``damped'' in [K].
\newline
{\bf 3.}
A typical example of an extremally small system $\Cal D $ is
a collection of the type
$\Cal D =\gb_{\ext}\ \&\ \{\gb_i \}$ where $\gb_{ext}$
is an ``external'' contour system (one has to be
careful with such a notion,
see below),
where
the ``internal''contour systems $\gb_i$ are {\it not\/} small
but the whole system $\Cal D $ {\it is\/} small.
The case $\{\gb_i\}=\emptyset$ is the most important
and common one, of
course.
\newline
{\bf 4.}
One should emphasize that there is always some freedom
in the definitions of these notions. For example the large
quantities $\tau, \tau' $
in the definition of a small resp. recolorable $\gb$ can
be changed;
they can also have an
`` individual value ''
$\tau_{\gb}$ ($\gg 1$) for any particular system $\gb$ etc.
These ambiguitites are more
important than the arbitrariness of the choice of $y$
in (3.49)
but they still have no ``physical'' meaning as they will {\it not}
affect
the physically meaningful notion of a stable phase used in Main Theorem.
\endremark
\proclaim{Proposition 3} Any
configuration $(x, \Cal D)$ ($\Cal D$ finite)
of a mixed model containing some strictly interior
small contour system contains at least one extremally small
contour system.
\endproclaim
\demo{Proof}(This statement will be used in the proof of Main Theorem
in a situation, when {\it several\/}, slightly different,
mixed models appear at different levels.)
Take a smallest (in $\prec$) strictly
interior small contour system $\gb$ in $\Cal D$.
If the collection of extremally small
(equivalently,
by induction,
small) contour systems which
are proper subsystems of $\gb$ would be empty then
$\gb$ itself would be extremally small !
\enddemo
As we yet announced, the
importance of the notion of a small resp. extremally small
contour system stems from the fact that extremally small contour systems
provide practically the {\it only
relevant example\/} of recolorable contour systems. Namely,
the following result
\footnote{One
can construct, of course, examples
of recolorable but not extremally small systems.
However, they correspond to some marginal cases only,
which are just not covered
by the particular choice of the constants
$\tau,\tau',\tau(\gb)$ but they are usually covered
by another, more suitable choice of these constants
in the definition above.} gives such a statement. It is a crucial step (together
with Theorem 5 above) in the proof of forthcoming
Main Theorem.
\proclaim{Theorem 6} If $\hat \tau = \tau'/ 12\nu $ then any
$ \tau'$--extremally small
contour system is $\hat \tau$--recolorable.
\endproclaim Theorem 6 will imply
(we mean the values from (2.14) and (3.34))
that after the completion of the recoloring procedures
of Theorem 5 (applied first to the original \ps \ model,
then successively to the partially expanded models
emerging from previous recoloring steps etc.), {\it no\/} small
contour systems $\gb\sqsubset \Lambda$ will be left in
the final mixed model in $\Lambda$.
This leads directly to Main Theorem, see section 8.
\endremark
\vskip1mm
\head {6. The proof of Theorem 6 }
\endhead
A contour system is $\gb'$ is called {\it tight\/} if it has
{\it no\/} contour subsystem $\gb$
satisfying
\footnote{Tightness of $\gb$ is the {\it same\/}
notion as the tightness
of its support $\supp \gb$ defined in section II.}
$$ \supp \gb \sqcap (\gb' \setminus \gb) =
\vv \cap \supp (\gb' \setminus \gb) =
\emptyset \ \ ( \Longleftrightarrow \ \
\vv \sqsubset (\supp (\gb' \setminus \gb)^c ) .\tag 3.52
$$
If the above relation {\it is\/} satisfied we say that $\gb$
is an {\it isolated\/} contour subsystem of $\gb'$.
Recall that we take the $l_{\infty}$ norm everyhere;
the
choice
of
$C=1$
instead of some other constant $C$ in $C \diam \supp \gb$
on the right hand side of (2.5)
is somehow arbitrary.
We will show that the proof of Theorem 6 can be reduced to the
case of extremally small {\it tight\/} systems $\gb$.
The smallness of $\gb$ will imply that
the quantity $A(\gb)$ in $F(\gb) = E(\gb)
-A(\gb)$ will become ``safely small'' with respect to
$E(\gb)$.
(The quantity $A^* (\gb)$ will be even {\it nonpositive\/}
in the most important case of the ``stable''
external colour of $\gb$ !)
The tightness of $\gb$ will then reduce
Theorem 6 to some {\it combinatorial statement\/}
relating the values $|\supp \gb|$ and $\conn \gb$.
See Theorem 7 below.
However, the case of systems $\gb$ which are
{\it not tight\/} is more
characteristic and more important for the proof. The quantities $A(\gb)$ (more
specifically, the quantities $A^*_{\text{max}}(\square)$)
will play rather important role in the announced
reduction to tight systems:
\definition { The skeleton }
\enddefinition
Let us denote the contour systems by symbols
$ \Cal D$ here, the symbol $\gb$ being now used mainly
for (pre)contours or even for cubes.
Consider the following
auxiliary construction in any volume $\Lambda$, for any
configuration $y$ on $\Lambda$ which is stratified,
at least locally in $\Lambda$. We will actually use it
below with the special
choice $\Lambda =
V_{\updownarrow}(\Cal D)\setminus \supp \Cal D $ (for
a contour system $\Cal D$ ) and with $y$ being the colour induced by
$\Cal D$ (it is only locally
from $\es$!) on $V(\Cal D)$.
\definition{Construction of the skeleton}
Given a stratified configuration $y$ and a cube $\square$
say that this cube $\square$
is {\it minimal nonsmall\/} if $A^*(\square , y) > \tau' ( \diam \square)$
holds for {\it some\/} extension
\footnote{ Here it would be perhaps more
convenient to have ``any'' instead of ``some''. Fortunately,
this subtlety could be important only for cubes
``very closely sticked to $\Cal D$'' and an inspection
shows that a slight increase of $E(\gb)$ in (2.14) (and therefore the increase
of $\tau' =
\tau -\varepsilon'$)
influences the lower bound for $F(\Cal D)$ more than all these
corrections.} of $y$, see (3.49) and no smaller cube which is at the same time a subcube
of $\square$ satisfies
such a condition. (Sometimes, for ``stable'' $y$ -- defined below in
Main Theorem such a cube $\square$ will {\it not\/} exist; however this is
not the case of a
typical nontrivial situation below.)
Given a volume $\Lambda$, let us now find
some smallest possible (in $\prec$) minimal nonsmall cube
$\square \sqsubset \Lambda$ (if there is some).
Take all the adjacent (having distance $1$ to $\square$) cubes
$\square'\ssubset \Lambda$ which are horizontal shifts of $\square$,
then take all the adjacent, horizontally
shifted cubes to the cubes just constructed etc.
Thus we obtain some ``layer'' (only partially filled in $\Lambda$;
cubes which would be too close
to $\Lambda^c$ are excluded !)
of cubes inside $\Lambda$.
Construct also other possible
partial layers {\it not intersecting\/} those constructed before,
according to the rule that a layer with a {\it smallest possible\/}
diameter of its
``paving blocks'' is constructed in each step. The exact meaning of the statement
that the layers would not mutually ``touch''
will be that the vertical distance between any two
adjacent layers is
bigger than the {\it logarithm\/} of the thickness
of both layers. There is of course some
arbitrariness in this requirement;
its reasonability will be seen below.
\footnote{ The reason is to keep $\Am$ roughly additive as a function
defined on the union of cubes of the skeleton.
We will see below that the logarithmic distance will assure this --
because
of the exponential decay of the terms $k_T$ in the sums (3.43)
used in the definition of the quantities $f_t$. See
Lemma below (3.54).}
The collection of all minimal cubes of $\Lambda$
thus constructed will be called the {\it skeleton\/ } of $\Lambda$.
We emphasise that the construction is applied in any
(generally nonstandard)
volume $\Lambda$ with a given {\it locally stratified\/}
configuration. In particular, a {\it skeleton\/} of the
``interior''$\Lambda =\vv \setminus \supp \gb$ of any
{\it extremally small system\/} $\gb$ is constructed
in this way.
\enddefinition
\remark{Note}
The fact that skeleton has a ``smallest possible
grain'' is slightly superfluous here but
it is useful not only below but also in other,
more detailed estimates used in the study of the completeness of
the phase picture constructed by Main Theorem.
Notice also that in the situation where
$\Lambda = \Lambda' \setminus \supp \Cal D$, $\Cal D$ extremally small,
the smallest possible grain of the skeleton of $\Cal D$
guarantees that the size of any strictly
interior (and therefore nonsmall)
$\gb \subset \Cal D$ is {\it at least as big\/} as
the distance from the nearest cube
(which has a {\it smaller\/}
diameter than $\gb$!) of the skeleton.
\footnote{ Take the covering rectangle $\square(\gb)$, denote
by $d$ its diameter. It follows from the construction
of the skeleton that a {\it cube\/} of a diameter
$d' \leq d$
must be already there, having a distance
at most $ d'$ from $\gb$.
Otherwise we could add
to the skeleton, at a suitable moment of its construction,
a cube of diameter $d$, touching the
covering rectangle of $\gb$.}
This observation is important
for establishing of the {\it tightness\/}
of the union of $\Cal D$ with its
skeleton.
\endremark
We conclude: for any extremally small $\Cal D$ in a volume $\Lambda$ with a
stratified boundary condition $y$ given on the
boundary of the complement of $\Lambda$,
we constructed the ``skeleton'' of the volume $
V_{\updownarrow}(\Cal D) \setminus \supp \Cal D $
which is composed of nonsmall {\it cubes\/}. These
cubes are ``densely''packed as formulated above
and the system $\Cal D$ enriched by the cubes of the skeleton
is tight.
\definition{Rearrangement of the (free) energy
inside of the skeleton of $\Cal D$}
\enddefinition
Let us define the following ``rearrangement''
of the energy inside of a given extremally small
contour system
$\Cal D$.
Imagine that the cubes of the skeleton will be treated
just as some new ``contours''.
The idea is to show that such an ``enrichened'' system $\Cal D^*$
of ``contours''
is {\it tight\/} in the sense formulated above
(and in (2.5))
and the value of its contour functional $F(\Cal D^*)$
(see (3.53) below)
is {\it smaller\/}
than
that of $F(\Cal D)$; however $\conn \Cal D^*$ is apparently
bigger than $\conn \Cal D$. Therefore, by checking the recolorability
of the enriched system we will also prove the recolorability
of the {\it original\/} system.
This will give the desired generalization of the
argument already given in (3.51) (given there for the
case when $\Cal D$, denoted as $\gb$,
is a single contour.
Let us show this in more detail:
Denote, as announced, by $\Cal D^*$ the collection of all contours of
$\Cal D$ and
also of {\it all the cubes\/} of the skeleton of $\Cal D$.
Let us make an
agreement that for any cube $\square$,
denoted here also as $\gb$,
we put (just to unify the notations
in the formula (3.54) below!) $|\supp \gb| = \diam \square$,
$A^*(\gb)=A^*(\square,x)$
and correspondingly
$$ F(\Cal D^*) = F(\Cal D)
+ \sum_{\square \in \Cal D} F(\square)
\ \ \text{where we define}\ \
F(\square) = \tau' \diam \square - A^*(\square,x) .\tag 3.53$$
Here, $x$ denotes
the configuration induced by $\Cal D$ on $\square$,
(extended somehow
to the whole $\zv$).
With this notation, using the Peierls condition (2.14) and the
definition (3.22) of $F(\Cal D)$ we can prove
(3.34) by showing the following inequalities:
Recall that
$$A^*(\square,x) \geq \tau' \diam \square \ \
\ \text{i.e.}\ \ F(\gb) \leq 0 \ \ \text{for} \ \gb =\square $$
(this is just the negation of (3.49)) holds for any
cube of the skeleton. Now we have the following relation between the contour functionals
of the original extremally small
system $\Cal D$ and the ``enriched''(by cubes of the skeleton)
system $\Cal D^*$: (The relation below
should be then written in the exponential form, to obtain
the bound (3.34*); see also (3.59 below!))
\proclaim{Proposition 1} We have the relation, using the
convention (3.53) and taking $\tau'$ as above
$$ \tau |\supp \Cal D| - A(\Cal D) \geq
\tau' |\supp \Cal D| - A^*(\Cal D)
\geq
\tau' |\supp \Cal D^*| - A^*(\Cal D^*) .
\tag 3.54 $$
\endproclaim
To prove this, notice that the first inequality just informs
about the approximation
of $A(\Cal D)$ by a slightly different quantity
$\Af(\Cal D)$ (Proposition 1 and 2 last section).
It is the {\it second\/} inequality which makes the core
of this Proposition; it will be shown now to
be a consequence of the {\it very definition of the skeleton\/}.
Write the negation to (3.49) as
$$ \tau' |\gb| - A^*(\gb) \leq 0 $$
for any new ``contour''
$\gb =\square$ of the skeleton of $\Cal D$.
The idea \footnote{What follows will be just a suitable
play with the quantities
$\Am|\square|$ and $\tau'|\diam \square|$ (where the first quantity
is replaced by the second one for any cube of the skeleton).
We can interpret this replacement as an
``installing of an artificial contour $\square$ ''.}
of the proof of (3.54) is
that the terms
$\tau'|\supp \Cal D| -A^*(\Cal D)$ are essentially {\it additive\/}
as functions
of the components $\gb$ of $\Cal D$. The
additivity of the energies $\tau |\supp \Cal D| =
\sum \tau |\supp \gb|$
(where the sum is over all contours of $\Cal D$)
is of course trivial.
Concerning the approximate additivity of the function
$A^*$
we have the following auxiliary result.
\footnote{The quantity $A(\gb)$ from section 3
is of course {\it exactly\/} additive for
disjoint volumes $\vv$ but it would have some other,
more severe disadvantages (than $A^*$) when the ``surface tension'' along
the vertical sides of the cubes will be discussed.}
\proclaim{Lemma (Approximate additivity of $A^*$)}
Let $\Cal D^* = \Cal C \cup \Cal S$
be a compatible collection of contours $\Cal C =\{\gb_i\}$
and mutually
disjoint cubes $\Cal S =\{\square_j\}$ such that the cubes from $\Cal S$
are not intersecting the contours of the system $\Cal C$.
Let the vertical distance between any two cubes from
$\Cal S$ whose projections to $\zet_{\nu -1}$ intersect
be greater or
equal than the logarithm of the diameter of the smaller cube.
Then (we take all the quantities $\Af$ with respect to the corresponding
external colour induced by $\Cal C$)
$$ \Af(\Cal D^*) = \Af(\Cal C) + \sum_{\square \in \Cal S}
\Af(\square ) + D(\Cal D^*) \tag 3.55
$$
where the correction term $D$ is small: It
satisfies the bound, with a
large $\tilde \tau(\tau)$
$$ |D(\Cal D^*)| \leq \sum_{\square \in \Cal S}( \diam \square)^{-\tilde
\tau} .$$ \endproclaim
\demo{Proof} Consider two such collections $\Cal D^*$
which differ just by {\it one
cube\/} $\square$ \ i.e. let we also have a
bigger collection \ $\Cal D^{**} = \Cal D^* \& \square $.
Assume that the cube $\square$ is not greater than any cube of $\Cal D^*$.
It is now sufficient to prove the bound
$$ |\Af(\Cal D^{**}) -\Af(\Cal D^*) -\Af(\square)|
\leq (\diam \square)^{- \tilde \tau}. \tag 3.55' $$
Notice that the quantity $f_t$ in (3.43) can be written also as the sum over intervals $I$
$$ f_t(y) = e_t(y) - \sum_{I \subset \zet\ : \ t
\in I} k_I $$
where $k_I$ is the sum of all the contributions to (3.43)
having a fixed projection $I$ of $T$ to the last coordinate axis.
Of course, we have a bound, for suitable large $\hat \tau$
$$|k_I| \leq \exp(-\hat \tau |I|).$$
Imagine that in the expression on the left hand
side of
(3.55') we {\it ignore\/} (when substituting these
quantities, for any vertical section of $\square$ and any $t$,
into (3.47)) {\it all\/} $I$ intersecting {\it both\/}
$\square$ and some {\it other\/} cube resp. contour of $ D^*$.
Then the relation (an analogy to (3.55'))
$$ A^*_{\text{ignore}}(\Cal D^{**}) - A^*_{\text{ignore}}(\Cal D^*)
- A^*_{\text{ignore}}(\square) = 0 $$
(we use the subscript``ignore'' instead of
``full'' to denote the quantities just mentioned)
is {\it exact\/}, as simple inspection shows.
The correction due to the quantities $k_I$ just ignored
is then obviously of the order $\exp(-\hat \tau d) $
where $d$ is the distance of $\square$ and the cubes from
$\Cal D^*$. This proves (3.55'), and therefore also
(3.55).
\enddemo
Let us continue now the investigation of the right hand side
of (3.54):
By (3.50),
the right hand side of the relation (3.54) is {\it greater\/}
than (we use here the very smallness
\footnote{ Notice that our use of the {\it squares\/} in the
definition of smallness is rather important technically.
Namely,
our method of the proof here relies quite heavily
on the fact that the enriched
(by cubes of the skeleton)
system $\Cal D^*$ of any small $\Cal D$ is again a
small system!
Our very definition of smallness using cubes
gives automatically such a property; otherwise much
more cumbersome (though possibly ``physically
more intuitive'')
notion of a smallness would be needed.} of the cube $\square(\Cal D^*)$ !)
$$ \tau'|\supp \Cal D^*| -\Af(\square(\Cal D^*))
\geq \tau' (|\supp \Cal D^*| - \diam \square(\Cal D^*))
\tag 3.56 $$
which is surely {\it greater\/} (notice that
$ \diam \square(\Cal D^*)= \diam \square(\Cal D)$!)
than, say (see the footnote below (3.59)), $\tau /2 \ |\supp \Cal D^*|$.
Thus it suffices to show now that the quantity $ \tau /2
\ |\supp \Cal D^*|$ (and, therefore, also the right hand side of
(3.56))
is greater than, say, \ $ \tau /12\nu \ \conn \Cal D$.
We will prove this by
proving the following result (Theorem 7).
Generalize again the notion of a tight system $\Cal D$
to any
subset of $\zv$ (and compare with (3.52)):
\definition{Tight sets}
\enddefinition
We will relax here slightly, for the purposes of the
forthcoming purely topological section,
our definition of
a tightness i.e. the relation $\sqcap$ on which the
notion of tightness of contour systems was based:
Now we will say that $S \subset T$ is isolated in $T$ (where
$ T \subset \zv $)
if we simply have
$$ \dist(S, T \setminus S) \geq \diam S. \tag 3.57$$
Say that $T$ is tight if there are no isolated subsets of $T$.
\remark{Notes}
1. The very concept of $ \vv$ above (starting from (2.16)) was
motivated by our introduction of the notion of a
{\it strict\/} interiority and diluteness.
Topologically, there is not much difference between
the notions of tightness of $\Cal D$ defined in terms of
tightness of {\it contours\/} of $\Cal D$
(as everywhere before,
which was more convenient in Theorems 3, 4, 5)
and tightness of $\supp \Cal D$ defined simply in terms of (3.57)
(which is slightly weaker requirement,
as simple inspection shows).
\newline
2. Notice that the system $\supp \gb^*$ is
already tight (even in the
sense of $\sqcap$)
because the definition of a skeleton of $\gb$
gives no room for isolated (in the sense of
empty $\sqcap$) subsystems
for the
enriched set $T = \supp \gb^* $ !
\endremark
\proclaim {Theorem 7} If $T$ is tight then
$$ \conn T \leq 6\nu |T| . \tag 3.58$$
\endproclaim
Apparently, the proof of this relation will finish
also the proof of Theorem 6. Namely, we can conclude the
arguments of (3.54) and (3.56) as follows:
$$ \tau |\supp \Cal D| - A(\Cal D) \geq \tau' |\supp
\Cal D^*| -
\Af(\Cal D^*)
\geq
\tau'|\supp \Cal D^*| - \tau' \diam \square(\Cal D^*)
\tag 3.59 $$
i.e. we finally get, using a bound
$|\supp \Cal D| > 2 \diam \square(\Cal D^*)$
\footnote{We omit the proof of this geometrical statement
(which is trivial for ``nonexotic'' systems $\Cal D$).}
and (3.58), the relation
$$ \tau |\supp \Cal D| - A(\Cal D)
\geq {\tau \over 2} |\supp \Cal D^*|
\geq {\tau \over 12\nu} \conn \Cal D \tag 3.59 $$
which gives the {\it recolorability\/} (3.34) of
the system $\Cal D$.
Let us now start the {\it proof of Theorem 7\/}
by restricting it first
the the case of sufficiently big cardinalities only.
(This assumption is used
in Lemma 1, section 7 below.) First
consider the case of sets of cardinality,
say, at most $1024 = 2^{10}$.
We will check
the bound (3.58)
for any tight set $T$ whose cardinality is
$d \leq 1024$.
More generally we will prove the following
weaker analogy of (3.58), which is valid for all
\footnote{We concentrate on $\nu =3$
and we do not try to optimize
all the estimates given here and below.} $d$:
$$ \conn \Cal D \leq 3
d \ \log_4 d \ ,\ \ \ d = \card \supp
\Cal D \tag 3.60$$
We will prove this by the induction, over the
number of connected components
$\{T_i\}$ ; $ i =1,\dots,n$ \
of the set $T$. The case $n=1$ is trivial.
Take the smallest component, say $T_1$,
of $T $ .
Notice that no more than $3 d_1$ unit steps
are necessary to connect $T_1$ with the rest
of $T$ by a connected path (having unit steps).
(We use here the fact that $T_1$ is {\it not\/}
isolated in $T$, compare (3.52), and assume first
for the simplicity
that $T\setminus T_1$ is tight, see below.)
Thus we have to check the following inequality,
where $d_2 $ denotes
the cardinality of $T \setminus T_1$:
$$ d_1 \ \log_4 d_1 + d_2 \ \log_4 d_2 +d_1
\leq (d_1 + d_2) \ \log_4 (d_1 + d_2).
$$
This inequality is proven by
elementary analytical calculation,
writing $d_1 = \alpha d_2$ where $\alpha \in (0,1)$.
If $T\setminus T_1$ consists of {\it several\/}
tight components $T_2,\dots,T_m$ then we need to connect
$T_1$ to each of them i.e. instead of the above
inequality we need a generalization
$$ (m-1) d_1 \ + \sum_{k \geq 1}d_k \ \log_4 d_k
\leq (\sum_{k \geq 1} d_k) \ \log_4 (\sum_{k \leq 1}
d_k)
$$ which is proven easily by induction over $m$
(recall that the cardinality $d_1 =\card T_1$
is assumed to be smallest from the numbers
$d_k =\card T_k$).
Thus it really suffices to prove the desired inequality
$ \conn T \leq 6 \nu|T| $
(see (3.58); we need it for $T = \gb^*$)
at the assumption that $\supp \gb^*$
has a cardinality
at least $1024$: \definition {Proof of (3.58)
for large $T$. Commensurately connected collections of cubes}
\enddefinition
It will be useful to {\it reformulate\/} the notion of a commensurately
connected graph to another language based on the employment
of {\it cubes\/} from $\zv$ instead of bonds
from $2^k \zv $:
\definition{Definition}
Say that the two cubes $\square , \square' \subset \zv $ are
commensurate if
$$ \square \cap \square' \ne \emptyset \ \ \ \text{and} \ \ \
|\log_2 \diam \square - \log_2 \diam \square' | \ \leq \ 2 .
\tag 3.61 $$
\enddefinition
(Notice that
the constant $2^1$ in (3.2) was replaced by $2^2$ in (3.61). This is
for purely technical reasons and will be convenient below.)
\proclaim {Proposition 2}
If \ $G$ is a commensurately connected {\it graph\/} then the collection
$\{\square(b), b \in G \}$ of covering {\it cubes\/} of bonds of $G$ is
commensurately connected in the sense above.
\endproclaim
This is immediate, by comparing (3.2) and (3.61).
The opposite relation (that any commensurately connected collection of
cubes can be ``approximated'' by a commensurately connected graph)
can be also established:
Introduce first another auxiliary geometrical notion.
\definition{Definition}
If $\square$ is a cube with a diameter
$2^k \leq \diam \square < 2^{k+1}$ then the lexicographically first
point of $\square \cap 2^k \zv$ will be called the anchor of
$\square$, denoted by $ a(\square)$.
\enddefinition
\proclaim {Proposition 3}
Let $\Cal S = \{\square_i\}$
be a commensurately connected collection of cubes from
$\zv$. Then there is a commensurately connected tree $\Cal T$
such that all the anchors $a(\square_i)$ are among the (possibly multiple)
vertices of
$\Cal T$ and
$$ |\Cal T| \leq 3 \nu |\Cal S| . \tag 3.62 $$
\endproclaim
\demo{Proof}
We may assume that $\Cal S$ is already a tree.
Take any commensurate bond $\{\square, \square'\} \ \in \Cal S$.
Write $[\log_2\diam\square] = k, [\log_2\diam \square'] = k' $;
we may assume that $k' \in \{k, k+1, k+2 \}$.
A straightforward inspection shows that it suffices to
consider the case $\nu = 1, k = 1$, and $ a(\square') = 0 $.
Notice that then the following path from
$a = a(\square) $ to $ a' = a(\square')$ can always be constructed :
\ \ $ a' = a + v_1 + v_2 + v_3 $ \ \
where the vectors $v_i$ having the lengths $2^{l_i} ;\ l_i \in \en$
satisfy the
following requirements:
$$ k \leq l_1 \leq k+1, \ l_1 -1 \leq l_2 \leq l_1 + 1 ,\
l_2 -1 \leq l_3 \leq l_2 +1 , \ k' \leq l_3 \leq k' + 1 .$$
It is clear that the tree defined by all the bonds
\ $\{a,a+v_1\},\{a+v_1,a+v_1+v_2\},\{a+v_1 +v_2, a'\}$
(where $a, a'$ vary over all commensurate pairs $\square,\square'$
and the triple above is repeated in the
direction
of any coordinate axis)
is commensurately connected.
\enddemo
\proclaim{Proposition 4}
Let $S \subset \zv $. Let $\conn_{\square}$ denote the cardinality
of a smallest commensurately connected collection of cubes
satisfying the following requirement: if all points of $S$ are added
(we identify the points of $\zv$ as cubes of diameter $1$) then the
whole collection is commensurately connected.
Then
$$ \conn_{\square} S \geq { 1 \over 3\nu} (\conn S). \tag 3.63 $$
\endproclaim
We will now prove Theorem 7 by showing that the inequality
$$ \conn_{\square} S \leq |S| \tag 3.64 $$
holds for any tight set $S$ whose components
have a cardinality at least $1024$.
\vskip1mm
\head 7. The Proof of (3.64)
\endhead
Let us make an agreement that an explicit choice of the appropriate
constants here will be given below only for the case of the dimension
$\nu = 3$. (Apparently, for $\nu > 3$ the final constant in
Theorem 7 is even better -- but we do not care here.)
Define a suitable collection of cubes
having a size $2^k $ where $k =1, 2, 3, \dots $ such that any cube in $ \zv $
can be ``packed'', with a reasonable ``accuracy'', by some
cube of the collection:
\definition{ Second covering cube }
Denote by $ \Cal K_k$ the collection of all cubes in $\zv$
which are shifts, by suitable values from the lattice $2^{k-1}\zv$,
of the unit cube $[0,2^k]^{\nu} $ in the lattice $2^k \zv $.
Write $ \Cal K = \cup_k \Cal K_k$ where
$k= 1,2,\dots $.
We have the natural ordering $\prec$ on $\Cal K$
extending the ordering by size resp. the lexicographic
ordering of shifts of one particular cube; this ordering
can be extended to suitable total
ordering of {\it all\/} cubes in $\zv$ which is in accordance
with the inclusion
relation as well as with the lexicographic order of mutually shifted cubes,
and
we denote by $\widehat{\square}$ the
(lexicographically first) cube from $\Cal K_k$, $k$ smallest possible,
containing $\square$. This will be called the
{\it second covering cube\/} of $\square$ resp. of a set $S$
such that $\square = \square(S)$.
\enddefinition
Notice the following fact :
if $\square$ is the covering cube of $S$ then the second covering
cube of $S$ contains $\square$ and has a diameter at most
four times bigger than $\square$.
\definition {Black and grey cubes of a set $ S \subset \zv$ }
Say that a cube $ \square$
is a black cube of $ S \subset \zv$
if $ \square \cap S $ contains at least $4 (\diam \square)^{1 \over 2}$
points, resp. at least $\diam \square$ points if its diameter is
smaller
than $16$. Any cube {\it from $\Cal K$\/} which is the second covering cube
of some black cube of $S$ will be called the grey cube of $S$.
(It obviously contains at least $2 (d')^{1\over 2}$ points
of $S$ where $d'\leq 4d $ denotes the diameter of the corresponding
grey resp. black cube.)
We will
show the following statements :
\enddefinition
\proclaim{Lemma 1}
If \ $T$ is tight and its connected components have a diameter at least
$16$ then the collection of its black cubes is commensurately
connected.
\endproclaim
\proclaim{Lemma 2}
The number of grey cubes of a size at least $1024$
of any set $S \subset \zv$ is no greater
than $1/2
\ |S|$.
\endproclaim
\proclaim{Lemma 3}
If $\Cal S = \{\square_i\}$ is a commensurately collected collection
(of black cubes of some set $S$) then the collection $\widehat{\Cal S} =
\{ \widehat{\square_i } \}$ (of grey cubes of $ S$)
is contained in some commensurately connected
collection $\Cal S'$ such that
$ |\Cal S'| < 2 |\widehat{\Cal S} |$.
\endproclaim
\demo{Proof of Lemma 1} We will proceed by the induction
over the number of points in $T$.
Say that $S$ is a {\it nice\/} subset of $T$ if it has the following property :
for any $t \in S$ there is a commensurately connected
collection of
black cubes $\{\square_i\}$ of $S$ which is concentric
(i.e. $t \in \square_i$ for
each $i$ ), starts in the covering cube of $S$ and ends in $t$.
Take some maximal nice subset $S$ of $T$. We will show that
$S =T$\ if \ $T$\ is tight.
Really, if $ N =T \setminus S $ is nonempty then either there is some
isolated subset $M$ of $N$ or $N$ is tight. In the former case
take $M$ as the smallest possible isolated (and therefore tight)
subset of $N$.
Then $M$ (or $N$ itself, in the latter case) is also nice by the induction
assumption.
Take the covering cube $\square(M)$ of $M$.
We claim that
there is some black cube $\square'$ of $S$
such that $$
\dist(\square(M),\square') < \diam \square(M)
\ \ \text{and} \ \ |
\log_2 \diam \square(M) - \log_2 \diam \square'| \leq 1
. \tag 3.65$$
This follows from the fact that $M$ can{\it not\/} be isolated
in $S\cup M$ (otherwise $M$ would be isolated also in $T$ ).
Therefore, there is some $t \in S$ whose distance from $\square(M)$
is no greater
than $\diam \square(M)$ and we take an appropriately large
black cube $\square' \ni t$. (Its existence is guaranteed
by the ``nicety'' of $S$.)
Now, if $\square^*$ is any black supercube of $\square'$
(black in $S$ ) then the supercube of $\square(M) \cup \square^*$
-- denoted as $\square^{**}$--will be shown to be
again a black cube (of the whole set $T$)
and this would mean that $S \cup N$ would be nice, as simple inspection
shows.(Check that there is now a commensurate
path from $t$ to $\square(M)$ and any commensurate chain
of cubes going ``up'' from $\square(M)$
through cubes of the type $\square^*$ can be
modified by going through
corresponding cubes $\square^{**}$. Thus $ S =T$. The modified
chain is clearly also a commensurate one, containing $\square(M)$.)
The observation that $\square^{**}$ is black in $T$ follows from the
following more general
\proclaim {Lemma 4} If \ $\square', \square'' $
($\square'' = \square(M) $ in the above application) are two
cubes which are black cubes of some
sets $ T' ,\ T'' \ ; \ T' \cap \ T'' = \emptyset$
and such that $\dist(\square',\square'') \leq
\diam \square'$ and
$ |\log_2 \diam \square'' - \log_2 \diam \square'| \leq 1$
then the covering cube $\square (\square' \cup \square'') $
is the black cube of the set $ T = T' \cup T'' $.
\endproclaim
The validity of Lemma 4 is easily seen (it suffices to consider the case of
the dimension $\nu = 1$!) from the inequality ($d$ denotes the distance
between two cubes $\square'$, $\square''$;
the diameter of $\square'$ is
assumed to be $1$ while the diameter of $\square''$ is
denoted by $x$)
$$
d \leq 1 \ \& \ 1/2 \leq x \leq 2
\Rightarrow (1 + x + d)^{1 \over 2} \leq 1 + x^{1 \over 2}.
\tag 3.66$$
\enddemo
\demo {Proof of Lemma 2}
We will give the proof only for the case $\nu = 3 $.
For the purposes of this proof modify the cubes $a + [0,2^k]^{\nu} $
from $\Cal K_k $ to the following form: $ a + [0, 2^k -1 ]^{\nu}$.
Then the system $\Cal K_k$ can be decomposed into $8$
pavings of $\zv$ by disjoint sets. Take the sum
$${8 \over 2}\sum_{10}^{\infty} {1 \over 2^{k \over 2}} < 7/16 \tag 3.70 $$
and imagine that any point of $t \in S $
transfers the ${1 \over 2} 2^{-k \over 2}$
-- th portion of its ``unit mass'' (3.70) to any cube of $\Cal K_k$
containing $t$.
By the definition of a grey cube,
the total mass thus transferred to any grey cube of $S$
is at least $1$
and, therefore, the cardinality of the set of all grey cubes of $S$
(of a diameter bigger than $1024$) is smaller than $7/16\ |S| \leq
1/2 \ |S|$.
\enddemo
\demo{Proof of Lemma 3}
This easily follows from the following observation :
if $\square, \square'$ are commensurate cubes then either
their second covering cubes
$\widehat{\square},\widehat{\square'}$ are also commensurate
or one of these latter two cubes can be replaced by an auxiliary,
twice bigger supercube from $\Cal K$ such that the first
statement is true.
\enddemo This concludes the proof of Theorem 7
if we moreover notice, that to
connect all the components of $S$ to their black supercubes
we can construct commensurate paths with less than, say, $1/2 \ |S|$ steps.
Thus, also Theorem 6 is proven.
\remark{Note (on the applicability of the notion
of a skeleton to a situation of \cite{Z})} Our
new Theorem 6 is a stronger and better replacement for the ``Main Lemma''
of [Z]. Really, one can use its (simpler) analogy also in the
translation
invariant situation of [Z]. Then it can have (e.g.) the following form:
If $\gb_i$ are mutually external ``large contour systems''
satisfying (in the notations
of \cite{Z}) the relation $a_q |V(\gb_i^q)| > \tau
|\supp \gb_i^q|$
then
$$ a_q |\ext| > \tau \Conn(\Lambda,\{\gb_i^q\})
\ \ \ \text{where}\ \ \ \
\ext = \Lambda \setminus \cup_i V(\gb_i),\tag 3.71 $$
the integer $\Conn(\Lambda,\{\gb_i^q\}$
on the right hand side denoting the cardinality
of a smallest possible set whose union with $\Lambda^c$ and
all $\supp \gb_i$ is {\it connected\/}.
With this lemma, one can rewrite the usual \ps \ theory
in a way analogous to that used here without an explicit construction
of the
contour models. See the lecture notes \cite{ZRO}
for more details.
\endremark
\vskip1mm
\head 8. Total Expansion of a General
\ps \ Model.
The Metastable Model. The Main Theorem.
\endhead
Up to now we worked with a general mixed model, though
having in mind that
the constructions developed above
should be used for a {\it study of successive expansions of
the original\/} \ps \ model.
To summarize our previous investigations:
Theorem 6 enables
to {\it repeat\/} the algorithm of {\it expansion\/} formulated
by Theorem 5, again and again.
\footnote{ One can start more generally with
a suitable mixed model: we will see that this remark
is important for some classes of {\it infinite range\/}
models.}
\definition{ Removable contour systems of a general mixed model}
A recolorable ($\hat \tau$ -- recolorable!)
contour system
$\gb$ which is a strictly interior contour subsystem of some
configuration $\Cal D$
will be called {\it removable\/}
if it is moreover smallest possible one in
$\prec \prec$. (We announced this notion
already in Section 5; recall that a basic example of a
$\hat \tau $-- recolorable system
is a $\tau'$ --{\it extremally small\/} one.) In a fixed volume $\Lambda$
we always accompany the above condition by the
requirement that $\gb$ would be also
isolated (see (3.52)) from $\Lambda^c $.
This is in accordance with the requirement
$\gb \sqsubset \Lambda$ used in our definition
of $Z_{\updownarrow}(\Lambda)$.
\enddefinition
\definition{Iterative use of Theorem 5. The totally expanded \ps \ model}
It is now important to realize that the
process of recoloring (of the collection of shifts of
a removable $\gb$) formulated by Theorem 5 can be applied
again to all the newly emerged
(after all their strictly interior
recolorable subsystems being ``swept'' by previous
applications of Theorem 5), removable
contour systems.
\footnote{Such new removable
systems appear as recolorable ``rudiments''
of contour systems
$\gb$ in the
situation
when all the recolorable subsystems $\gb' \subsetneqq
\gb$ were
already expanded by the previous applications
of Theorem 5. Notice that in the course of this
successive construction, due to the appearance of the
new quantities $k_T(y)$, occasionaly
some previously recolorable (but too big to be removed
at that moment)
objects may turn out later (at the moment they {\it could\/}
be
removed)
to be nonrecolorable. Also, some previously nonrecolorable
objects $\gb$ can be recolorable later from the point of view
of some later expanded model-- but we do
{\it not\/} remove them yet, to avoid confusion
in the construction of clusters $T$ (using relation
$\prec$). These
marginal phenomena do not affect
the sense of our contruction.
Let us stress once again that only the
``smallest possible
recolorable contour systems'' are recolored
at any stage of the expansion.}
This iterative process
can be therefore repeated
up to the very moment
when there are {\it no\/} removable, equivalently
{\it no\/} ($\hat \tau$--)recolorable, in particular
{\it no ($\tau'$--)extremally
small\/} (by Theorem 6)
strictly interior
contour subsystems
in the
``final'' mixed model, where
``where nothing recolorable remains''. This
final mixed model will be called
the
{\it totally
expanded \ps \ model\/} corresponding
to the abstract
Pirogov -- Sinai model given by Hamiltonian (2.12)
and satisfying (2.14).
\enddefinition
\definition{$ G$ -- expanded \ps \ model}
This is the result of successive
applications of Theorem 5 (see above)
performed up to the moment
when no $G$ --removable
\footnote{Some (marginal cases of) newly $G$--recolorable
contour systems
$\gb$
which were $G'$--nonrecolorable,
$G' \subset G$ in previous expansion steps may appear, as we
noted above. We do {\it not\/} recolor them yet.}
$\gb$, $\supp \gb \prec G$ remains. (This operation
is done simultaneously in all finite $\Lambda$,
for any boundary condition $y \in \es$.)
The following result is then a direct
consequence of the very construction of the totally
expanded model.
It is of a crucial importance in the following.
\enddefinition
\proclaim{Corollary (of the definition of a totally
expanded model)}
In the totally expanded model, only those
contour subsystems remain which are not recolorable
(therefore: not small)
and
contain
no recolorable contour subsystem.
\endproclaim
\definition{Residual systems}
A contour system $\gb$ which exist as a configuration
of the totally expanded model (equivalently,
$\supp \gb$ -- expanded model)
\footnote{ Residuality is decided on the level
of $\supp \gb$ -- expanded model, and later
expansion do not matter!}
will be called a
{\it residual\/} one. If \ $\tilde \Cal D$ is a contour
system in $\Lambda$, under some
(unstable) boundary condition $z \in \es$
remaining from $\Cal D$ after the total expansion of the model
we will say that $\tilde \Cal D$ is a {\it residuum
of \/} $\Cal D$.
\enddefinition
\remark{Notes}
1. An equivalent characterization of residuality in terms
of the related notion of a metastability will be given below.
Notice that
the removability in a volume $\Lambda$ of $\gb$ means
just the recolorability of $\gb$
\& {\it residuality\/}
of all the isolated contour {\it sub\/}systems of $\gb$
\& isolation of $\gb$ from $\Lambda^c$ if we are
in a volume $\Lambda$ .
Speaking about recolorability, residuality etc.
of some $\gb$ in Corollary
above one has
in mind the corresponding property of $\gb$ in the
``provisional'' mixed model, constructed up to the moment
when these properties of
$\gb$ are checked (just before applying
Theorem 5 in the case of removability). In this
provisional mixed model, there are no more any
removable systems $\gb',\gb' \prec \prec \gb$.
2. In fact, (non)recolorability
or (non)smallness of a contour system $\gb$
in the
final, totally expanded model and in the provisional one
mean almost the same. (However,
only the {\it latter\/} notion is used!)
More precisely we note that for cubes $\square$ of a comparable (or smaller)
size as
$\gb$, the quantities $A^*(\gb)$ and $A^*(\square)$
are almost
the same for both the provisional (expanded ``up to the size of $\gb$'')
and the final expanded model. (Notice that the quantities
$A(\gb')$ are {\it exactly\/}
the same, in both mixed models, for contour systems
satisfying the condition $\diam \gb' \leq \diam \gb$.)
More precisely, clusters $T$ having a {\it different\/} value
in considered mixed models (the provisional one and the final one)
have a size {\it at least\/} $\diam \gb$
and thus the difference between the corresponding values of
$A(\gb)$ discussed above is of the order
$\varepsilon^{\diam \gb}$; a
tiny quantity compared to $\tau |\supp \gb|$!
This fact, together with the fact that that $A^*$ and $A$ are almost the same
in any mixed model,
will be used later in the proof of Main Theorem
(relation (3.81))
and also in (3.90) and below it.
\endremark
\definition{Metastable model}
A configuration $(x_{\Cal D}^{\text{best}},\Cal D) $
which is $y$ -- diluted,
\ i.e. equal to
$y \in \Cal S$ outside of some set having standard
finite components, will be called
{\it $y$ -- th metastable\/}, shortly metastable,
if there is a sequence of successive removals
(expansions by Theorem 5)
of removable subsystems (of the remainders of \ $\Cal D$\
left by the preceding applications of Theorem 5)
leaving an {\it empty set\/} at last.
\footnote{ In contrary to the usual applications
of the \ps \ theory we do {\it not\/} try
to express such a property in terms of
``external contours'' of the system. Such a characterization
would be cumbersome.}
The restriction of the
original abstract Pirogov -- Sinai model (with the Hamiltonian (2.12))
to all $y$ -- th metastable configurations
will be called the {\it $y$ -- th metastable model\/}.
If the abstract model was constructed as a representation
of an original model (2.2), then we define the
$y$ -- th metastable submodel
of (2.2) as the restriction of the Hamiltonian (2.2) to the
configuration space $\ex^y_{\text{meta}}$ of all diluted configurations
$x \in \ex$ whose representations $x \equiv (x_{\Cal D}^{\text{best}},\Cal D)$ in terms
of contours are $y$--th metastable in the sense above.
We can also say that $\Cal D$
is {\it residual\/} if and only if it has {\it no\/} isolated
metastable
subsystems. Now we are able to formulate our
{\it basic result\/}: \enddefinition
\definition{The free energy of the expanded model}
Recall the quantities $f_t(y)$, $y\in\Cal S$
which were defined in (3.43) for {\it any\/} abstract
mixed model.
Consider now these quantities for the case of the totally expanded model
whose construction (starting from
the original \ps \ model) was just described.
In such a case, the quantity $f_t(y)$ will be denoted by
\footnote{To emphasise the fact that $k_T(y)$ are
not simply just some additional small quantities added to $e_t$
and $E(\gb)$ but some
{\it very special functions\/}
of the original abstract \ps \ model.}
$$
h_t(y) \equiv \ f_t(y) . \tag 3.72
$$
\enddefinition
\definition{Ground states of the totally expanded model:
minimalization of
$\{h_t(\cdot)\}$}
Say that $y \in \es$ is a ground state of
the totally expanded model if the inequality
($[t]$ denotes the collection of
all $t'= t+
(0,0,\dots,0,t_{\nu}); \ t_{\nu} \in \zet $)
\footnote{Obviously, this
sum does not depend on the choice of $t \in \zv$.}
$$
\sum_{t'\in[\,t\,]}(h_{t'}(\tilde y)-h_{t'}(y) ) \geq 0
\tag 3.73 $$
holds for any $\tilde y \in \es$ which
differs from $y$ on a layer of a finite
width only. \enddefinition
\definition{Stable elements of $\Cal S$}
A stratified configuration $y\in\Cal S$ for which there is
{\it no\/} residual contour system $\gb$ such that
$(x_{\gb})^{\ext}= y$ will be
called {\it stable\/}.
In other words, $y$ is stable if the collection of
configurations $ x\neq y$ having the value $y$ outside $\Lambda$
is empty in the totally expanded model, for {\it any\/}
finite $\Lambda$.
This notion will be used only
under additional
(very natural, see the discussion below (3.94))
assumption that contours of an {\it arbitrarily large
diameter\/} exist in the original \ps \
model, for any external colour
$y \in \es$.
\enddefinition
\proclaim{Main Theorem} Consider an abstract
Pirogov -- Sinai model (section II, 6)
defined
by Hamiltonian (2.12), (2.14), with a
large $\tau$.
Moreover, assume that contours of
all sizes exist, for any external colour $y \in \es$.
Then the stable configurations
$y \in \es$ are precisely
those configurations from $\es$ which are the
ground states of $ h_{t}(\cdot) $.
\footnote{The quantities $h_t(y)$
can be interpreted as
the densities, at the point $t \in \zv$, of the free energy of the
corresponding $y$-- th metastable model. To understand
their meaning, see e.g. (2.26).}
Moreover, for any stable $y $ there exists an ``abstract Gibbs
measure'' $P^y$
on the configuration space
$$X^{\text{abstract}}_{\Cal I,y}
\subset X^{\text{abstract}}_{\text{fin}} \tag 3.74$$ (see
Section II, 7)
of the given abstract \ps\ model
whose almost all
configurations are $y$--th metastable.
The
conditional finite volume
probabilities $P^y_{\Lambda}$, $y$ stable,
conditioning of $P^y$ being taken with respect to
strictly $y$--diluted configurations in $\Lambda$,
correspond to the Gibbs probabilities
on the
ensembles $X^{\text{abstract}}_{\Lambda,y}$.
\endproclaim
For the Hamiltonian (2.2), we get the following straightforward
consequence, discussed in more detail
in Section 9 (see the formula (3.93) and the discussion
of more general boundary conditions below it) of the paper.
\proclaim{Corollary} If the abstract Pirogov -- Sinai
model of Main Theorem
represents some ``physical'' model given by Hamiltonian (2.2),
then for any stable $y $ there exists a Gibbs
state of the model (2.2) on $\ex = S^{\zv}$ whose almost all
configurations
can be identified,
in the contour representation $x_{\Lambda}
= (x^{\text{best}},\gb)$, with a suitable
element of
$ X^{\text{abstract, meta}}_{\Cal I,y}$
of all $\Cal I$ -- finite,
$y$--th metastable configurations.
\footnote{One could prove also the exponential decay of correlations in
any (metastable) Gibbs state thus constructed. We omit
the details. See, however, Section 9
for some information explaining, or at least preparing
the
ground (relation (3.93)) for the proof
of
these facts.}
\endproclaim
\remark{Notes}
{\bf 0. } Clearly, the families of configurations
$\ex_{\text{meta}}^y$ are mutually disjoint for different $y \in \es$.
We will not study here in much detail the structure of a typical
configuration of the ``$q$ -- th Gibbs state''. See, however,
the final section 9 for some information.
(This problem deserves a detailed treatment. However, it seems
reasonable to do this in connection with an
investigation of other related questions, like
the completeness of the phase picture constructed here.
We plan to devote a separate paper to these questions.)
\newline {\bf 1. }
There are no other stratified Gibbs states of such an abstract
model. We are not giving here the proof of such a
completeness of our phase picture (characterized by the
stable values of $y \in \Cal S $).
It can be done similarly as in \cite{Z}.
See also some comments in the section
Concluding Notes below. However, we plan a more systematic treatment
of this and related
questions in a separate paper.\newline
{\bf 2. }
By the phrase that ``the $y$ -- th Gibbs state
can be identified with the
corresponding $y$ -- th metastable model'' we mean that
almost all configurations of this Gibbs state are $y$ -- diluted
and moreover the ``islands''(say the components of $\vv$
where $\gb $ is the collection of all contours of the
the considered configuration
$x$; $\vv$ the set of all points of $x$ which are
{\it not\/} $y$ -- stratified) are typically ``small''
and ``rare'', but distributed with a uniform density
throughout $\zv$. For a more
complete statement, see the section 9 below.
\newline {\bf 3.}
Having in mind that the quantities $h_t(y)$ can be
effectively computed from the expansions (3.43)
(within a given precision; of course in full this is a
horribly complicated sum -- but its terms are converging
{\it very\/} quickly) our Main Theorem gives, in fact, a
{\it constructive criterion\/} for finding the stable
values $y\in\Cal S$.
Practically, one may suggest an approximate
finding of stable values of $y$ from some
auxiliary ``$M$--expanded''
model ($M$ is some square, for example)
where only those removable contours whose size
does {\it not\/}
exceed the size of $M$ are recolored.
In fact, even for squares $M$ quite small some
useful approximations can be found, often enabling yet
to distinguish
between the stable and nonstable $y$. This is because the series (3.43)
are really very quickly converging and moreover we often have some
additional symmetry in the special cases of interest;
like the
$+/-$ symmetry in some special cases of the Blume Capel models.
For those models, even the smallest size 2$\times$2 of the square $M$
can be useful -- see \cite{BS}. Namely, considering only
first few terms in (3.43) a correct conclusion
about the stability of the $0$ phase resp.
the $+/-$ phase can be made.
\newline
{\bf 4.}
In fact, in ``reasonably shaped'' (see below)
finite volumes $\Lambda $ there is no noticeable
difference between the behaviour of the stable $y$ --th
phase and another $\tilde y$ --th phase if $$
a=\sum_{t\in[\,t\,]}(h_{t}(\tilde y)-h_{t}(y) ) \tag 3.75
$$
satisfies, say, \ $\tau a^{-1} >(\diam \Lambda)^{\nu-2} $. Quite straightforward
estimate of the quantities
$$
A(\square) \leq a |p(\square))| \leq
a (\diam \square)^{\nu- 1} \leq
\tau \diam \square
\tag 3.76
$$
where $p$ denotes the orthogonal projection on $\zw\subset
\zv$ shows that the equations of the type
(3.49) cannot be violated in such a small
volume.
``Small'' can have a meaning ``having a diameter of the order
$10^{27}$'' here. Namely, consider the situation
(quite typical in applications)
when we have some apriori given subset
$\es' \subset \es $
of ``all the reasonable candidates for
almost ground states'' -- such that
we are sure that no stable $y$ could be found outside of
$\es'$. In such a case the size of the quantity $a$,
more generally the size of the difference between
various $\sum_{t= (t_1,\dots,t_{\nu -1},(\cdot)} h_t(y),
\ y \in \es'$ is only of the order $a_G +a_P
\diam \Lambda$
where $a_G$ is the maximal deviation between
the ``ground''sums
(over the vertical lines
$\{t= (t_1,\dots,t_{\nu -1},(\cdot))\}$)
$\sum_{t= (t_1,\dots,t_{\nu -1},(\cdot))}
( e_t(y)$, $y \in \es'$
while
$a_P$ is the maximum of the difference
between values on the left hand side of the
exponential Peierls bound (2.14)
given for the
same $G =\supp \gb$
but for the different external colours $y \in \es'$.
If we take e.g. a perturbed Ising model with
a horizontally invariant
external field whose ground vertical sums do not
exceed a value $\lambda$
then one can estimate that
$a_P$ is of the order $\exp(-n \beta J) \lambda$\
(where $n > 4 $ or even more)
and apparently one does not need to have $\lambda$
very small to reach that, say $a_P < 10^{-100}$
for {\it small\/} temperatures $\beta^{-1}$. So the
question of stability in such a volume is often just
a question of behaviour of the
``ground'' part of $h_t(y)$
i.e. of finding of minimum of the
vertical sum of $e_t(y)$, within a given precision.
\newline
{\bf 5.}
Though the question of the existence of {\it at least one\/}
stable $y\in \Cal S$ is not the absolutely crucial one (as the
preceding note shows) one should mention, nevertheless,
that at least one stable $y\in\Cal S $ really {\it does exist\/}:
Say that $y\in\Cal S $ is $N$-ground if for any $\tilde
y\in \Cal S $ such that $\tilde y=y$ for $|t_\nu|\geq N$
we have the inequality (3.73).
Now, if the configurations $y^{N}\in\Cal S $ are $N$-ground
then a suitable subsequence of $\{y^{N}\}$ must converge to some
ground (``stable'') value $y\in\Cal S $. (We use the compactness
of the space $\Cal S $ in this argument as well as the quick convergence
of the cluster series for the quantities $h_t(y)$).
\newline
{\bf 6.}
For some models, like the Ising model with stratified random
external field, the collection of all (almost) stable
$y\in\Cal S$ can be {\it very rich\/} and the phase diagram
-- as the function of all (vertically dependent) values
of the field -- extremely complicated. We plan to
study this particular case in some later paper.
\newline {\bf 7.}
The latter example case shows that it is not very reasonable to try
to formulate
results about the shape of the {\it phase diagram\/} in full generality here
-- because possibly infinite parameters are present in
the Hamiltonians of the
stratified type.
However, if we call by a phase diagram of the model the very {\it mapping\/}
$$ \{ \ \ y \in \Cal S \ \longmapsto \{ h_t(y) \} \ \ \} \tag 3.79 $$
then the information about the actual phase diagram, its dependence
on the parameters in the Hamiltonian etc. can be deduced from
(3.79) ; just by using suitable variants of the implicit function theorem
(possibly with infinitely many variables). However, this
paper is not
on analysis of manifolds and so we omit these questions completely.
It is worth noticing here that, in order to get a best possible
smoothness of the mapping (3.79) (and of the mappings derived
from it by implicit function theorems),
it may be reasonable to {\it modify\/} suitably
the definitions of extremally
small contours etc. -- to obtain the best available differentiability
(even local analyticity) properties of this mapping. This question
also deserves a separate study, like in
\cite{ZA}.
\endremark
\demo{Proof of Main Theorem}
The key statements are Theorems
5 and 6 above which guarantee
the existence of the totally expanded model
with {\it no\/} small contour subsystems.
In fact, these two theorems opened the possibility
of the very {\it formulation\/} of our result
based on the construction of the
quantities $h_t(y)$ having rather strong and
useful properties.
Noticing this, one has to add only a few additional
observations.
1) If $y$ is a ground state,
i.e. it satisfies (3.73),
then no contour system $\gb$
such that $(x_{\gb})^{\ext}=y$ is residual. Really,
we have from the
definition of $A^*_{G}(\square, y)$, $\square =\square(\gb)$
(the subscript $G =\supp \gb$
denotes the $G$ -- expanded model (3.47))
the inequality, with a very small
$\tilde \varepsilon$ (of the order
$(\varepsilon^*)^{\diam
\square}$, see below for $\varepsilon^*$)
$$ A^*_G(\square, y) \leq \tilde \varepsilon \diam \square.
\tag 3.80 $$
One could take, from (3.73), even {\it zero\/} on the right hand side
of this relation if the appropriate {\it totally\/}
expanded model
would be taken here.
So we have to use first an estimate
(similar in its nature to (3.42))
$$|A^*_G (\square,y) -
A^*_{\text{totally expanded}} (\square,y)| \leq
(\varepsilon^*)^{\diam \square} \diam \square $$
where $\varepsilon^* = C \varepsilon$.
Thus we see, that the smallness of any contour
system $\gb$
with the external colour $y$ satisfying (3.73)
is (having in mind the -- just mentioned --
slight difference betweeen different mixed models
which could be considered there) automatically
fulfilled (actually with a very {\it small\/}
constant $\tilde \varepsilon$
instead of $\tau'$ !)
and therefore, under such boundary condition $y$, there is {\it no\/}
difference between the original \ps\ model and the metastable one.
Thus any contour system $\Cal D$ with boundary condition
$y$ is small. Existence of such systems clearly implies
also an existence of such systems with
connected $\vvd$. These systems are, however,
extremally small, because they are small and
all their subsystems are residual.
\footnote{More precisely, nonrecolorability of subsystems
of $\Cal D$ is guaranteed in a slightly different mixed
model (than the totally expanded one) only. This means that
the constants like $\varepsilon'$ in the proof of Theorem 6
have to be slightly changed and our choice of $\tau'$
must be slightly more careful than we argued before.}
By Theorem 6, they are recolorable
and this is a contradiction with
the definition of the fully expanded model.
2) On the other hand, if
$y$ is {\it not\/} a ground state then we will
show that some residual systems
having the external colour $y$ {\it do exist\/}:
Namely, then there is some
$\tilde y \in \es$ differing from $y$ only on
some layer $L$ of a finite
width, say $d$, such that the vertical sum below
(it does not depend on $t$) is
$$
\sum_{t'\in[t]}h_{t'}(y)-h_{t'}(\tilde y) \geq \delta
\tag 3.82 $$
for a suitable $\delta >0$ . Take a very large box $B\subset
\zw$ such that $$
\tau |\partial B| << \delta |B|
\tag 3.83
$$
and consider the volume, having the form of a ``desk''
$
\Lambda = \{t\in\tilde L, \hat t \in B\}
$
where
$\hat t$ denotes the projection
to $\zw$ and $\tilde L$ is another, thicker layer containing
the above layer $L$
``sufficiently inside itself''. The
thickness $2d$ would be
amply sufficient; the purpose of the thicker layer is to
control all the nonnegligible (with respect to $\delta$)
cluster terms acting
in the difference $\sum (h_t(y) -h_t(\tilde y))$.
Take the configuration $x=y$ outside of $\Lambda$,
$x=\tilde y$ inside $\Lambda$. Then, if we compute the
quantity $A^*(\Lambda)$ for the volume $ \Lambda $,
we have according to (3.47) the folowing bound from below:
$$
A^*(\Lambda) = \sum_{t\in\Lambda}(h_t(y)-h_t(\tilde y))
\geq
(\delta -\epsilon)|B| \gg
\tau |\partial B| \ \ \text{i.e.} \ \gg
\tau|\diam \Lambda| ,
\tag 3.84 $$
which shows
that $\Lambda$ is not a small volume
i.e. some residual contours do exist !
This argument is absolutely straightforward for the original spin model
(2.2) where contours of almost arbitrary shape can be
found. For an abstract \ps \ model the above conclusion
(on the existence of residual contours under
unstable boundary conditions)
requires in fact the assumption
that contours
having
arbitrarily big diameters
really {\it do exist\/}, in the abstract model,
for any external colour $y \in \es$.
\footnote{ This is of course fulfilled for abstract
\ps \ models arising from (2.2). One
could define artificial models {\it not\/}
allowing such big contours; then a
``one sided'' modification of the statement
of Main Theorem must be made.}
3)
Moreover one has, under stable boundary conditions
$y$, quickly converging expansions
of partition functions in any volume, and this implies the
validity, as we explain below in (3.93), of the properties of the
``$y$ -- th Gibbs state''$P^y$ stated in the Main Theorem.
\enddemo
\definition{Generalization to quickly decaying infinite
range Hamiltonians. An outline}
\enddefinition
In Corollary, end of section II
we formulated the concept of
an essentially one dimensional interaction and outlined
how to work with it. This concept
is just a realization of the idea to
``pack'' all the interactions $\Phi_A$ resp.
cluster expansion terms (the latter can be interpreted
as some additional interactions, too)
into
corresponding (as small as possible) cubes.
It is important to assume a quick decay of the
long range interactions. Otherwise there would be problems
with establishing of the Peierls condition
for the ``aggregates'' defined below.
If we inspect all the technical steps we made
during our development of the expansion process in part III
then the conclusion is that
there are the following two groups of problems caused
by additional infinite range (sufficiently
quickly
decaying) interactions :
1) Let us stick first to the formulation given in terms
of the abstract \ps \ model (as always up to now;
below we will briefly mention
also the case of spin models developed in \cite{EMZ}).)
Notice that instead of an abstract \ps \
model we can already start with a nontrivial
{\it mixed model\/} as well. Actually, our Theorems
3 and 4 and all the treatment of the sections
2 - 7 is already given in a form prepared for such a
generalization --
assuming that the original, long range, essentially
one dimensional interactions fit our assumptions
(3.1) on the given mixed model.
We have to explain what (3.1) would mean
for essentially one dimensional interactions
$\Phi_{\square}$:
Define $\conn \square$ as\ $C \log \diam \square$\
for a suitable $C > 1$; then apparently
only a decay with a sufficiently big power of distance
is required, so that the machinery of
sections 2 -- 7 could be applied without a
change. \newline
2) Few comments about the {\it spin\/} models
(used in the applications made in
\cite{EMZ}):
If we have a nontrivial additional long range interaction
$H^{\omega,\varepsilon}$
in the situation of the short range model (2.2)
then the following auxiliary construction has to be made,
to adapt the spin case to the case of abstract
\ps \ models studied so far.
Given the perturbative interactions
$\Phi_{\square}^{\omega,\varepsilon}(x)$ which now
act also
{\it inside\/} of the contours (assuming that the
restriction
of a given configuration to a cube is stratified)
compute the
quantities
$e_t(y)$ as in (2.8), taking in account also
the {\it long range interactions\/} $\Phi_{\square}$.
Of course, we cannot adjust
the contour energies $E(\gb)$ to fulfil
(2.12) exactly in the case of infinite
interactions. Therefore, let us try to express
the necessary corrections as additional (weak!)
{\it interactions
acting on cubes intersecting contours\/} only:
Define
$$\Phi^{\text{corr}}_{\square}(x_{\square})
=\sum_y \xi_{\square}^{x,y} \Phi_{\square}^
{\varepsilon,\omega}
(y_{\square}) \tag 3.85$$
for any such cube $\square$, where the sum
is over all stratified $y$' s which are equal to
$x$ at some stratified point of $x$ belonging to
$\square$. (Typically, the sum contains only {\it one\/}
term.) The coefficient $ \xi < 1$ is computed such that
the expression (2.12) would be
{\it exact\/} with the
addition of the interactions
$\Phi^{\text{corr}}_{\square}(x_{\square})$.
(This is clearly possible, we omit formulas analogous to
(2.11) for $\Phi^{\text{corr}}_{\square}(x_{\square})$.)
These correction potentials can now be treated by the
following
``high temperature expansion'':
Given a configuration $x$ take the product over the
family $\Cal C(x)$ of all
cubes intersecting, but not included in,
the union $B$ of supports of
all contours of $x$. (Contours are still
defined with respect to the
``unperturbed'', short range Hamiltonian $H_0$!) $$
\prod_{\square \in \Cal C(x)} \exp(-\Phi_{\square}^{\text{corr}}
(x_{\square})) =
\prod_{\square \in \Cal C(x)} (1+ k_{\square}^{\text{corr}}
(x_{\square})) =
\sum_{\Cal S} \prod_{\square \in \Cal S} k_{\square}^{
\text{corr}} (x_{\square})) \tag 3.86$$
where the sum is over all subsystems $\Cal S \subset
\Cal C(x)$ of
the collection of cubes above
and where
$ k_{\square}^{\text{corr}} = \exp(-\Phi_{\square}^{\text{
corr}})- 1$ decay like $\Phi_{\square}$.
Write $ k_{\square}^{\text{corr}}(x_{\square})$
also in the
exponential form $$ k_{\square}^{\text{corr}} (x_{\square})
= \exp(-\Phi_{\square}^{\text{new}}(x_{\square}))
$$
and perform the corresponding
expansion of the right hand side
of the expression
$$\prod_{\square \in \Cal C(x)} \exp(-\Phi_{\square}^{\text{corr}}
(x_{\square})) \prod_{\{\gb_i\}}\exp(- H_{\text{ref}}
(\{\gb_i\})) \tag 3.87 $$
where $H_{\text{ref}}$ denotes the ``reference'' short
range
hamiltonian (2.12) but with
the {\it long range interactions incorporated into the
definition of\/} $e_t(y) $ (not $E(\gb)$ !)
as mentioned above. Then a new
abstract \ps \ model with the same densities $e_t(y)$
as above
is obtained, if we define the {\it new
contours\/} of the {\it new abstract \ps \ model\/} as
{\it conglomerates\/} of the original contours
and ``high temperature cubes'' \ $\square$ \
from the product above. The new conglomerates
play the role of usual contours of the setting (2.12)
and their energy is the sum of the Peierls energies of
original contours
contained in the conglomerate and of the energies
$\Phi^{\text{new}}_{\square} (x_{\square})$.
We may now check the Peierls condition
of the type (2.14), interpreting cubes
of diameter $n$ as suitable commensurate collections
of $C \log n$ points (in order to minimize the entropy
estimates for the conglomerates produced
by (3.87)) for
the aggregates of the cubes from the above expansion
and contours of the configuration $\{\gb_i\}$.
We do not study this case in a detail yet
but notice that an exponential decay of the type
(2.28) is sufficient even for $\omega$ close to 1
if $\varepsilon$ in (2.28) is sufficiently small.
We mention that the case of infinite range interactions in
the usual translation invariant
\ps \ theory was first systematically treated
by papers \cite{YMP}.
\vskip1mm
\head 9. Properties of Typical Configurations, Gibbs States in finite Volumes
\endhead
We are still in the situation of an abstract Pirogov -- Sinai model
(and back in the case of finite range Hamiltonians).
Given $y \in \es$ denote by $X^y_{\text{meta}}$ the configuration space
of all configurations of the $y$ -th metastable model.
More precisely denote by $X^y_{\text{meta}}(\Lambda)$ the configuration space
of all configurations $x_{\Lambda}=(x_{\Lambda}^{\text{best}}, \Cal D) $
such that $y $ is the external colour of $x_{\Lambda}^{\text{best}}$
and
$\Cal D$ is strictly diluted in $\Lambda$
i.e. $\Cal D \ssubset \Lambda$.
We have the probability
measure $P^y_{\text{meta},\Lambda}$ on $X^y_{\text{meta}}(\Lambda)$,
the corresponding partition function $Z_{\text{meta}}(\Lambda,y)$
being given by summation of
$\exp(-H(x_{\Lambda}|y))$ over all $x_{\Lambda} \in
X^y_{\text{meta}}$. The
Gibbs factor $H(x_{\Lambda}|y)$ is given by (2.12).
Write $P_{\Lambda}^{y}$ instead of
$P^y_{\text{meta},\Lambda}$ if $y$ is stable.
Having defined these ``strictly diluted'' Gibbs conditioned probabilities
$P^y_{\text{meta},\Lambda}$
the question now is whether some limit
$$ P^y_{\text{meta}}(\Cal E )
= \lim_{\Lambda \to \zv} P_{\text{meta},\Lambda}^y(\Cal E)
\tag 3.88 $$
exists for a sufficiently rich collection of events
$\Cal E$.
``Sufficiently rich'' would mean, in the {\it spin\/} model,
e.g. the collection of all cylindrical events i.e. events
depending on a finite number of coordinates.
In the abstract \ps\
model, a simplest example of such an event is $\Cal E(t)$
below
in (3.89).
The limit then gives for $y$ {\it stable\/}, a prescription
for a
{\it Gibbs measure\/} $P^y$ on the whole configuration
space $\ex$. (More precisely
prescription for the probabilities of some special
events like below. We omit here the full proof of the fact that $P^y$ given by formulas below
really gives a uniquely defined $\sigma$ --
additive probability measure.)
For nonstable $y$ this limit
can be interpreted as the ``metastable
Gibbs measure'', denoted by $P^y_{\text{meta}}$.
Let us show for example that the limit
$$ P^y(\Cal E (t)) = \lim_{\Lambda \to \zv}
P_{\Lambda}^{y}(\Cal E(t)) \tag 3.89
$$
exists for any stratified
$y$ and
for any $t \in \zv$ if the event $\Cal E(t)$ is
defined as follows: \
``$ x \in \Cal E(t) \ \ \text{iff} \ \
t \notin \vvd$ where $\Cal D$ is the
collection of all contours of the configuration
$x$''.
(For nonstable $y$ we have an analogous
statement for $P^y_{\text{meta}}$.)
This event will be called
``$t$ is strictly {\it exterior\/} point of the configuration $x$''.
Notice that for each $\Lambda$ we have the formula
(and analogously in a general metastable situation)
$$P_{\Lambda}^y(\Cal E(t)) =
(Z^y_{\updownarrow}(\Lambda))^{-1} Z^y_{\updownarrow}
(\Lambda \setminus t)
\exp(-e_t(y)). \tag 3.90 $$
We can expand, in the totally expanded model,
both the partition functions
on the right hand side of (3.90). The possibility of such an expansion
follows from the very notion of stability of $y$.
We have
the
following expression
(analogous expansions will be commented below also
for some more general boundary conditions):
$$
\log P^y_{\Lambda}(\Cal E(t)) =
\sum_{T : \ T \cap \cap (\Lambda^c
\cup \{t\}) = \{t\}} k_T(y) \tag 3.91 $$
where the quantities $k_T(y)$
are from the expansion (3.1) of
$Z^y_{\updownarrow}(\Lambda)$ resp. of
$Z^y_{\updownarrow}(\Lambda\setminus t)$ and
the sum is over those (quickly decaying !) quantities
$k_T$
only which
``touch'' $t$ in the sense
that $T$ does {\it not\/}
satisfy the condition $T \ssubset \Lambda \setminus t$
\ but does satisfy the condition $T \ssubset \Lambda$.
Clearly, then we have ($\varepsilon$ is
from
(3.1) and $C =C(\nu)$)
an approximate relation
$$ 1 - P^y_{\Lambda}(\Cal E(t)) \ \leq C \ \varepsilon . \tag 3.92 $$
This suggests that the ``islands'' of almost
any configuration $x_{\Lambda}= (x_{\Lambda}^{\text{best}},\Cal D)$
(see below) are really
mostly ``small sized and rare''
because they do not usually intersect a given (arbitrarily chosen)
point $t \in \zv$.
To make this intuitive description of almost any
configuration
(which is quite characteristic for the Pirogov -- Sinai
theory and for
the phase picture this theory gives)
more technically more
accessible, we will define below also the
``frames'' of islands $(x,\Cal D)$ in a
way grasping also some important features (namely
the appearance of residual contours) of the regime
appearing {\it inside of $\vvd$\/} and thus allowing
also {\it expansion formulas\/}
for the event ``a given frame appears''.
\definition{External contours, islands}
Define the relation
$ \gb \to \gb' $ whenever
($ \supp \gb \prec \supp \gb'$
and moreover)
$\vv \subset V_{\updownarrow}(\gb') $.
(We consider {\it no shifts\/} of $\gb'$ here!)
Let us interpret the relation $\gb \to \gb'$
by saying that contour
$\gb$ is ``inside'' of the contour $\gb'$. Say
\footnote{One should not take this characterization
too literally.
Of course the usual intuitive concept of ``exteriority''
fits
the scheme above but also some contours $\gb$
which are ``mostly outside'' of $\gb'$ satisfy the
condition above.
However, in contrary to the concept of admissibility which will be
{\it fundamental\/} in what follows the relation
``$\to$''will {\it not\/} be employed below except of
some notes at the very {\it end\/} of the paper, when
explaining the meaning of our Main Theorem.}
that $\gb$ is an external contour of an finite
contour system
$\Cal D$ if $\gb \to \gb'$ for no contour
$\gb'$ of $\Cal D$. Clearly, $ \cup \Sb \gb \in \Cal D \endSb \vv =
\cup \Sb \gb \in \Cal D^{\text{ext}} \endSb
\vv $
where the second union is over all external
contours of $\Cal D$.
Any connected component $I$ of $\vvd$ (equivalently,
one can take $\Cal D^{\text{ext}}$ here)
will be called an {\it island\/} of $\Cal D$.
Denote by $\Cal D_{I}= \{\gb \in \Cal D: \vv \subset I\}$,
analogously define $\Cal D^{\text{ext}}_{I}$.
\footnote{ Maybe ``archipelago of contours''
would be an appropriate name.
Typically, however, islands are mutually
external, simply connected sets and
for islands with ``holes'',
the external colour of the island penetrates also inside the
holes (Proposition below). So, {\it any\/} island is
effectively an ``external one''.}\enddefinition
\proclaim{Proposition}
For each island $I$ of
any configuration, one can
determine the ``colour'' $y = x_I=
x_{\Cal D_I} \in \es $ of
the island (i.e. of the system $ \Cal D_I$)
which appears
also inside of the
(possible) interior components of $(V_{\updownarrow}(
\Cal D_I))^c$.
Any contour of $\Cal D$ either belongs to $\Cal D_I$
or is isolated from it.\endproclaim
\demo{Proof}This is a geometrical statement
saying that regions with different ``interior'' colours
(which can appear ``inside'' of the contours of
the island $I$)
are already covered by $V_{\updownarrow}(\Cal D_I)$.
This is clear
if $\Cal D_I$ is a single contour (then
obviously $V(\Cal D) \subset V_{\updownarrow}(\Cal D)$).
Otherwise some
additional considerations are needed, we omit them here.
\enddemo
\definition{Frames of a configuration, the probability
$P^y_{\Lambda}[\gb]$}
A contour subsystem $\gb \subset \Cal D; $ of some
island $\Cal D_I$ of $\Cal D$
will be called a {\it frame\/},
more precisely frame of the island $\Cal D_I$),
if it contains all the
external contours of the system $\Cal D_I$
and moreover
the system $\Cal D_I \setminus \gb$ is strongly
diluted in $(\supp \gb)^c$
and {\it metastable\/}. (In other words, frame of
$\Cal D_I$ is formed by $\Cal D_I^{\text{ext}}$
and the {\it residuum\/} of $\Cal D_I \setminus
\Cal D^{\text{ext}}_{I}$.)
Denote by $P^y_{\Lambda}[\gb]$ the probability (in
$P^y_{\Lambda}$; below
we are mainly interested in the stable
values of $x_{\gb}^{\text{ext}}= y$ and in the
thermodynamic limit $\Lambda \to \zv$)
of the
event ``$\gb$ is a frame of a given configuration''.
The following theorem is then an analogy of the fundamental relation
(3.23). \enddefinition
\remark{Note} Above, metastability is meant in the
local colours induced by
$\gb$ outside of $\supp \gb$. We are able
to decide whether $\gb' \subset \Cal D \setminus
\gb$ is recolorable or metastable, because the covering cube
of $\gb'$ is still contained in $(\supp \gb)^c$.
The case of a ``small'' system $\gb$ such that
$\vv \setminus \supp \gb$ does not contain other contours
and everything what can happen outside of $\vv$ is (meta)stable
is the most characteristic one, of course,
and $\gb$ is commonly either a single contour or a collection of
one external contour and some interior residual
ones. \endremark
\proclaim{Theorem 8. Expression for $P^y_{\Lambda}[\gb]$
}
We have the formula, for any contour system
$\gb \ssubset \Lambda$
(this is a more detailed version
of (2.24); \
$y$ denotes the external colour of $\gb$)
$$ P^y_{\Lambda}[\gb] = \exp (-F_{\text{full}}(\gb)) \ \exp(\sum_{
T\cap \cap \Lambda^c \ne \emptyset}^{T \cap \cap \supp
\gb \ne \emptyset}
k_T^{\text{cor}}(\gb) )
\tag 3.93 $$
where $F_{\text{full}}(\gb) =
F_{\text{full}}^{\infty}(\gb)$ is like
in the definition (3.22), the superscript $\infty$
indicating that we take the totally expanded,
not the $G$--expanded, $G =\supp \gb$
(like in Theorem 3)
model when
constructing the quantity $F^{\infty}_{\text{full}}(\gb) =
E(\gb)-A^{\infty}_{\text{full}}(\gb) $.
\footnote{In contrary to Theorem 3,
we {\it do not want \/} to
expand here the frame $\gb$.
We do not even assume a Peierls type inequality for
$F_{\text{full}}(\gb)$. However, then it
can happen (for nonstable colour $x_{\gb}^{\text{ext}}$
only !) that the statement (3.94) below
is violated. (The difference between
$F^{\infty}$ and $F^{G}$ plays almost no role in it.)}
The quantities
$k_T^{\text{cor}}(\gb)= k_T(x_{\gb}) -k_T(x_{\gb}^{\text{ext}})$
represent, in the notations
analogous to that of
(3.23), just the difference
$F_{\text{full}}^{\infty}(\gb)
- F^{\infty}_{ \Lambda}(\gb)$; the
sum over $T$ being empty
in the thermodynamic limit.
\footnote{ A formula similar to (3.93) (and closer to
the situation
of (3.23)) could be obtained when taking \ $F_{\text{full}}^G (\gb),
\ G =\supp \gb$
there. However, then it would describe the
{\it conditional probability\/} that the
frame $\gb$ appears under a condition that
all the bigger contour systems $\Cal D, \ G \prec \supp \Cal D$ are
forbidden ! }
\endproclaim
\remark{Notes}
1. Notice that $F_{\infty}$ is a horizontally translation
invariant (nonlocal !) quantity.\newline
2. The concept of a frame is our true substitution
for the usual concept of an external contour (the
latter
being {\it not\/} so important here).
Notice that ``to be an frame'' is {\it not\/}
a cylindrical event. However, this is an event
``close'' to a cylindrical one
and it can
be expressed much more nicely, by the formula (3.93),
than the purely cylindrical events.
Also, the purely cylindrical events like $
\Cal E_{\gb} =\{ \Cal D:
\gb \subset \Cal D\}$
can be expressed
in terms of
``frame events'', by specifying the very
frame $\gb' \subset \Cal D$
to which $\gb$ either
{\it belongs\/} or (we consider here, for simplicity,
only
these two ``extremal'' situations)
which {\it controls\/} $\gb$ in the sense
that $\vv \subset V_{\updownarrow}(\gb')$.
In both cases we have cluster expansion formulas
based on (3.93) for these events; more direct formulas
(just (3.93))
are obtained in the former case.
\footnote{ The probability of $\Cal E_{\gb}$ is then the sum
of the probabilities of different (``connected'', by Theorem 6)
frames $\gb', \gb' \owns \gb$ -- for which we have (3.93) and,
therefore, for stable $y$,
also
(3.94)!}
In the latter case,
we have the formula (3.93) for the probability of any
frame $\gb'$ ``controlling'' $\gb$ and
{\it then\/} we have usual
formulas for the {\it conditional probability\/}
of the appearance of $\gb$ under $\gb'$.
\footnote{ We do not formulate here these conditional
probabilities
(of the more detailed events happening in the
``interior''
$V_{\updownarrow}(\gb') \setminus \supp \gb' $ of some
frame $\gb'$).
This is already a straightforward
task, using the properties of
conditional Gibbs distributions.}
\endremark
\demo{An outline of the proof of Theorem 8}
This is just another application of the method
of (3.23) (more precisely (3.25)),
for $\Cal D = \emptyset$.
Expand
(by suitable modification of Main Theorem applied
only to the
volume $(\supp \gb)^c$ under a condition $\gb$
(not to the whole lattice $\zv$!)
the partition function
of the event
``$\gb$ is a frame, with external
colour $y$, of a configuration
$\Cal D$ in $\Lambda$'' and divide it by the expansion
of the whole partition function $Z_{\updownarrow}(\Lambda)$.
Then (3.93) is obtained, with a value of
$F_{\text{full}}(\gb)$, which is {\it very\/} close
to the value $F(\gb)$ introduced in section 3.
The difference from Theorem 3 is that
here we do {\it not\/} recolor $\gb$ (even if it could
be
recolored at the moment
when the recoloring proces ``reaches the level of $\supp
\gb$'')
and then continue the recoloring process
only in the set $\zv \setminus
\supp \gb$. \footnote {Instead of recoloring
in the whole $\zv$. Namely, the method of Theorems 3,4,5
can be extended to any, even irregular, ``lattice''
if the statement on horizontal invariance of
the result is rephrased properly. The possibility
to use
boundary conditions which are only locally from $\es$
was yet footnoted in Theorem 3.} One uses
an analogous (but not exactly the same) arguments as
those
used there. Recall that
the totally expanded model used in the
definition
of $F^{\infty}(\gb)$ employs also
clusters containing contours
``bigger than $\gb$'' (but isolated from it).
In fact,
the difference between the values of
$F^G(\gb), G=\supp \gb$ (used in section 3)
and $F^{\infty}(\gb)$ (used here)
is of the order
$ \varepsilon^{\diam \gb}$ only. It is given
by contributions
$k_T$ of clusters $T$ containing elements bigger (in
$\prec$)
than
$\supp \gb$.
\enddemo
\proclaim{Corollary of Theorem 8} If a frame
$\gb$ is recolorable
then
$P^y[\gb]$ satisfies
the estimate,
with $\varepsilon = \exp(-\hat \tau)$ and $\hat \tau$
slightly smaller than
$\tau / 12\nu$ ,
$$ P^y[\gb] \leq \varepsilon^{\conn \gb}\ . \tag
3.94 $$
The probability of the event ``there is a frame
around $t$
having a diameter $\leq d$'' can be estimated,
for $y$ stable,
by $\text{const} \cdot \varepsilon^d$.
The mean relative area occupied by the supports
of such frames $\gb$ resp. by the
volumes $\vv$ of almost any configuration
is then equal, in the Van Hove limit, to some $\varepsilon' =
\varepsilon'(\varepsilon) $ resp. some
$\varepsilon'' = \varepsilon''(\varepsilon) $
independently of the particular
choice of the configuration.
There is an exponential decay of correlations in the probability
$P^y$ and $P^y$ has support in
the countable union of compact sets
$X^{\text{abstract}}_{\tilde I}$, $\tilde I$ shift of $I$.
\endproclaim We do not prove these (rather straightforward
once (3.93) was established) facts here.
\remark{Note}
The formulas like (3.93) only {\it suggest\/} that
$P^y$ exists really a
{\it probability measure\/} on $\ex$ with the properties stated above.
Also, to gain a control over some more general diluted
or ``almost diluted'' (see below in last section
for
natural examples of
more general boundary conditions which
are less restrictive than the
strictly diluted ones) partition function
and finally to interpret the measure $P^y$
even as a Gibbs measure on $\ex$
i.e. as a Gibbs measure of the {\it original Hamiltonian\/}
(2.2) some additional, more or less straightforward
work is needed. See below for some comments.
\endremark
\definition{Stratified Gibbs States in Arbitrary
Finite Volumes}
\enddefinition
Finally we give here a short information about the
properties of Gibbs ensembles in finite volumes
arising from some
more general boundary conditions
than the ``strictly diluted'' stratified ones which
we used everywhere
before.
Our aim here is not to investigate the
partition functions under
arbitrary boundary
conditions in
{\it full generality\/}
(this question and the related question
of the completeness of the phase picture constructed
by Main Theorem
deserve a separate study)
but only to study some noncomplicated and quite
natural boundary conditions which are still, in a sense,
``near'' to the requirement of the strict
$y$ -- diluteness in $\Lambda$, $y \in \es$
but are formulated in a simpler and more traditional
way {\it not\/}
based on the notion of tightness (and isolation)
developed in this paper.
It will be convenient to remain still in the framework
of the abstract \ps \ situation. Namely, even the context
of (2.2) with a stratified
$x_{\Lambda^c} = y_{\Lambda^c},\ y \in \es$
can be described in the language of (2.12),
assuming now however that
contours of the spin configurations
$x_{\Lambda} \cup y_{\Lambda^c}$
{\it may even intersect\/}
the set $\Lambda^c$ (but apparently
only at the condition that they also intersect $\Lambda$).
Denote by $X^{\text{gen}}(\Lambda, y)$ the collection
of all such (more general than strictly diluted) configurations of the abstract
\ps \ model and by $P^{\text{gen}}_{\Lambda,y}$ the corresponding
Gibbs probability on this ensemble. Such a broader
concept
includes also the traditional partition functions
(2.4) if
$x_{\Lambda^c}$ equals to some $y_{\Lambda^c},\ y \in \es$.
\definition{On some more general, $y$ -- like and
``fuzzy'' boundary conditions}\enddefinition
One can generalize the arguments given below even to some
more
general $y$ -- like boundary conditions on
$\partial \Lambda$.
We mean the boundary condition $x$ given on $\Lambda^c$
such that $x =y$ holds
for ``most'' points of $\partial \Lambda^c$. One should be
a little bit careful
what does mean the requirement ``for most points of
$\partial \Lambda$''. However, we will not
discuss literally such a generalization here.
In fact, it is more instructive to study, for {\it any\/}
volume $\Lambda$, the following ``fuzzy''
boundary conditions given by some {\it additional
Hamiltonian\/} acting ``outside of the volume
$\Lambda$ but in the vicinity of $\partial \Lambda$''
(say in some layer $\partial_r \Lambda^c$)
which ``supresses sufficientlly all the deviations
from the horizontally constant $y$ --th regime''
with the help, say, of additional
chemical potential ``disfavouring
everything else than pure $y$''.
It can be shown that such a ``fuzzy''
boundary
condition
gives the same phase in
the thermodynamic limit (for reasonable (conoidal)
volumes) as
the strictly diluted $y$ -- th boundary
condition studied before, if $y$ is stable.
Far from
$\partial \Lambda$ one will not see any noticeable
difference between those
boundary conditions. Formulas like (3.93) enable
to prove these statements quite easily.
\footnote{
More precisely the action of any
such boundary condition
on some $\partial_r \Lambda^c$ can be modelled with the help
of a suitable
{\it strictly\/} diluted boundary condition
in a {\it suitable larger volume\/}
$\tilde\Lambda$ with $\dist(\Lambda,\tilde \Lambda^c)
> r$,
however with some
additional energies (denoted by
$\tilde E, \tilde e$)
added (for ``fuzzy''boundary conditions)
to the original energies (2.12).
The added energies are such that
$\tilde E(\gb) \geq 0$ and
$\tilde e_t(z)\geq 0 ,\ \tilde e_t(y) = 0,\
z \in \es$ and they act
only in $\tilde \Lambda \setminus \Lambda$.
We could formulate everything in such a more general
way already starting from (2.12)
if all the additional requirements above
could be considered
as a ``weak preturbation of pure $y$''.
The only change would be
that the quantities $k_T(z),\ z \in \es$,
lose their horizontal invariance
in the case when $T$ touches $\tilde \Lambda
\setminus \Lambda$ and
also $e_t(z), z \in \es, y \ne z$
are increased in the
strip $\tilde \Lambda \setminus \Lambda$.
Then, a short look at formulas like
(2.26), (3.91), (3.93)
shows that the conclusions like the
stability of the $y$ -- th phase
constructed by these more general boundary conditions
in conoidal volumes $\Lambda$
(see below), and also in the
infinite
volume limit {\it remain valid\/}
if we keep $r$ in $\partial_r \Lambda^c$
fixed and not very
big (depending on the value $\tau$).
The case of boundary conditions
which are `'far from pure $y$'' is, however, a
more complicated problem.}
\definition{On shapes of $\Lambda$ with
negligible entropic
repulsion/attraction
effects}
\enddefinition
Concerning the shape of the
considered volumes $\Lambda$, one has to be
a little bit more careful.
Namely for volumes $\Lambda$ whose boundary contains huge
planar parts (like volumes having a shape of a `` desk'',
or even cubes!)
the boundary term representing the
``entropic repulsion or attraction''
can be important.
In fact one can compute these effects from relations
like (3.93) (more thorough study
is prepared for the paper \cite{HMZ}) and the
stability of $y$ can lose its validity near the boundaries
of such volumes,
if more general boundary conditions
than the strictly diluted ones (which were,
among other properties, ``designed
not to
feel these subtleties'')
are considered.
Namely, if the configuration in $\Lambda$ is not
strictly diluted,
$\vv \cap \Lambda^c \ne \emptyset$
then $A^{\infty}_{\text{full}}$ from Theorem 8
can change its value significantly if $V(\gb)$ sticks
to the flat part of $\partial\Lambda$.
The corresponding generalization of (3.93)
then uses the quantity
$$
A_{\Lambda}^{\infty}(\gb,x|\vv) =
\sum_{t\in V(\gb)}(e_t(y)- e_t(x)) +
\sum\Sb T\ssubset \vv\setminus \supp\gb \endSb
^{T \ssubset \Lambda} k_T(x)-
\sum_{T\ssubset \vv}^{T\ssubset \Lambda}k_T(y)
\tag 3.95
$$ (where $x = x_{\gb}$) in the fully expanded model and this
can be written as (compare (2.24)!)
$$
A_{\Lambda}^{\infty}(\gb,x|\vv) =
\sum_{t\in V(\gb)}(h_t(y)- h_t(x)) +
\Delta(\gb)
\tag 3.96
$$ where $\Delta(\gb)$
(we have to check that this quantity
``does not erode'' $E(\gb)$) is given by
$$
\sum_{t\in V(\gb)}(e_t(y)-h_t(y)- e_t(x) +h_t(x))
+
\sum^{T\ssubset \Lambda}\Sb T\ssubset \vv \setminus \supp \gb \endSb
k_T(x)-
\sum_{T\ssubset \vv}^{T\ssubset \Lambda}k_T(y).
\tag 3.97 $$
This {\it can\/} be a dangerous quantity if the
``upper (lower) ceiling of $\Lambda$ is very close to the
upper (lower)
ceiling of $V(\gb)$, because then the last two sums give
a term proportionate to the area of the flat part
of $\vv \cap \Lambda$ !
\footnote{ This cannot happen in conoidal volumes
where the last sum is of the order $\supp \gb$
only.}
and in such a way, the stability of $y$ can be {\it
destroyed\/}
(or, on the contrary, established for some {\it
nonstable\/} $y$)
in these volumes which are ``very flat from below resp.
from above''. Then, some residual frames
$\gb$
marking the shift to some other, more stable (in $\Lambda$ !)
horizontally invariant regime $\tilde y$
can appear (only in {\it huge\/} volumes $\Lambda$
and only near their boundary).
In order to control these effects
we have to introduce some {\it additional\/}
(rather mild)
assumptions on the ``shape'' of $\Lambda$,
like that there
are ``no flat ceelings in
$\partial \Lambda$''.
(This will assure that $A^{\infty}_{\Lambda}(\gb)$ is rather
benign, not eroding $E(\gb)$).
The following definition gives a characteristic example
of such a volume.
\definition{Conoidal volumes}
Say that a volume $ \Lambda$ is a conoidal volume (or conoid)
if it contains, with each ``horizontal set'' $ B \subset
\zv_m \cap \Lambda$, where $\zv_m$ is the collection of points of $\zv$
with the fixed last coordinate $t_{\nu}=m$, also
the whole ``cone'' $\{t \in \zv : \dist(t,B) \leq C_{\dist(t,
\partial B)}\}$ where $ C \log r < C_r < r$ holds
with not too small constant $C$.
\footnote{ The name ``cone'' is appropriate, of course,
only if $C_r \approx r$ (like $C_r =r/2$).
$\partial B$ is taken in $\zv_m$.}
\enddefinition
Now, in conoidal volumes
\footnote{ So, parallelpipeds are slightly
awkward volumes
in principle. One has to be a little bit
careful what stability
means, especially near their boundaries, and this is a rather subtle question.
Also, some ``almost stable'' $y \in \es$ can look like stable in suitable
large parallelopipeds.
On the contrary, volumes
like cones or octagons are trivial in this respect
and no entropy repulsion/attraction
effects are observed for them.} $\Lambda$
it is rather clear
that $A^{\infty}_{\Lambda}(\gb)$ is very close to its
``normal'' value $A^G(\gb)$ (and $A^{\infty}(\gb)$)
and any reasonable $y$ -like boundary condition
(e.g the ``fuzzy'' one)
gives a very similar picture as the strongly diluted
$y$ --th one.
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\by L. Chayes, E. Olivieri, F. Nardi, M. Zahradn\'{\i}k
\paper Low temperature phases of
some two dimensional ``unisotropic'' spin models
\jour in preparation
\endref
\medskip
\ref\key ZRO
\by M. Zahradn\'{\i}k
\paper A short course on the Pirogov -- Sinai theory
\jour Lecture notes of the lectures held in Rome II, February --
March 1996
\publ preprint Univeristy Roma II, 1996
\endref
\medskip
\ref \key LMS
\by J. L. Lebowitz, A E. Mazel, Yu. Suhov
\paper An Ising interface between two walls:
Competition between two tendencies
\jour preprint Cambridge
\yr Jan. 1996
\endref
\medskip
\ref \key LM
\by J. L. Lebowitz, A E. Mazel
\paper A remark on the low temperature behaviour of the SOS interface
in halfspace
\jour preprint Cambridge
\yr 1996
\endref
\medskip
\ref \key YMP
\by Yong Moon Park
\paper Extension of Pirogov - Sinai theory to
infinite range interactions I,II
\jour CMP, 114
\yr 1988
\pages 187 - 242
\endref
\medskip
\ref \key EMZ
\by A. Van Enter, J. Miekisz, M. Zahradn\'\i k
\paper Nonperiodic long range order at positive temperatures
\jour preprint
in preparation
\yr 1997
\endref
\medskip
\ref \key PECH
\by E. A. Pecherski
\paper The Peierls condition is not always satisfied
\jour Select. Math. Soviet 3
\yr 1983/84
\pages 87 -- 92
\endref
\medskip
\ref \key MIE
\by J. Miekisz
\paper Classical lattice gas model with unique
non degenerate but unstable periodic ground state
\jour Comm. Math. Phys 111
\yr 1987
\pages 533 -- 538
\endref
\medskip
\ref \key HMZ
\by P. Holick\'y,M. Zahradn\'\i k, A. Messager
\paper Entropy repulsion layers in
Blume Capel and Potts models
\jour in preparation
\endref
\medskip
\bye
ENDBODY