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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\Gammab{\boldsymbol\Gamma}
\def\gb{{\boldsymbol\Gamma}}
\def\vv{{V_{\updownarrow}(\gb)}}
\def\zv{\Bbb Z^{\nu}}
\def\zw{\Bbb Z^{\nu-1}}
\def\ps{Pirogov -- Sinai}
\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}}
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\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 \v{c} . 202/96/0731
and \v{c}. 96/272.
\endthanks
\date June 17, 1996
\enddate
\keywords Low temperature Gibbs states, stratified
hamiltonians
and phases, 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
\cite{S}, \cite{Z}.
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 having translation noninvariant
structure with a ``rigid interface'' goes back to the pioneering
Dobrushin's paper \cite{1}.
Several authors continued this study; we note e.g\. articles
\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 (between the
two translation invariant phases above and below) 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.
Recently, 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 \cite{HZ}.
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{MZ} 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
simplifications 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 depth.
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 as a very important
{\it testing quantity\/} (allowing one to decide whether the contour
is ``small'' or not) but instead of the construction of auxiliary contour
models a central point of our approach is the idea of a
succesive partial {\it expansion\/} of the model based
on an important new
technical step which is called {\it recoloring\/} of the 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 yet as a contour .
We will show that
the ``metastable'' submodels of the given model
(constructed for any stratified boundary
condition) can be {\it expanded completely\/}
and that for the ``stable'' boundary conditions, the corresponding
metastable model will
be identical with the original ``physical'' model.
Roughly speaking, all the
external contours will be ``small'' resp. ``recolorable''
(in the sense of
\cite{Z} resp. of this paper) in such a situation. The organization
of our expansions will make unnecessary estimates like
``Main Lemma'' of \cite{Z}.
Instead, we have now a more powerful method based on our Theorems 5
and 6.
\newline
To summarize, we converted the Pirogov -- Sinai theory just to a carefully
organized {\it method\/} of (succesive)
{\it expansion of some 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 also interpreted as
``walls'' (the terminology of \cite{D}) of the interface and the events
happening ``inside'' resp\. ``outside'' of the wall cannot
be tracted as 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 the testing quantity $F(\gb)$
called
the ``contour functional'' of a given contour $\gb$ 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. aggregates).
Let us mention some typical examples which can be treated
by our method.
\roster
\item 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 weeting layer of minuses appear.
\newline
b) Models in layers (like in \cite{MS}, \cite{MDS})
\item a) Models of the Blume-Capel type with spins belonging to some
finite set $Q\in\er$ and with the hamiltonian consisting of
a quadratic (e.g.) 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 the
approximately same depth). 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.
\newline
b) Such a situation appears, in the Fortuin -- Kasteleyn
representation, also
for the Potts model with large number of spins
below the critical temperature, where phases of the type
order -- layer of disorder -- another order exist. (The
paper \cite{HMZ} which is under preparation will be devoted to these
questions.)
\item
a) ``Sedimentary Ising rock''. 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
$\Bbb Z^{\nu-1} x\{0\}$ 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 rich structure of (many) 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 ``horizontally invariant'' external
field with approximately zero mean over the vertical
shifts, one should expect phases with infinitely many
layers (of changing $\pm$ phases), and the problem is to
compute the exact positions of the layers.
\newline
b) We will see later that 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.
\endroster
Our main result is given in part III, section 8:
\newline
In the translation invariant Pirogov -- Sinai theory, one
constructs, for any reference configuration (i.e. for
any ``local ground state'')
a quantity
called the ``metastable free energy''. If the minimum of this quantity
is attained in 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 of the Ising type whose configurations
correspond to various ``horizontally invariant regimes''
of the original model. The
{\it ground states } of this one dimensional model
correspond to the different {\it stratified} Gibbs states
which we are looking for!
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 in principle computable as they
are given by cluster expansion series (with complicated, but
very quickly decaying terms). In the case
when $y$ is the ground state of the corresponding
one dimensional model (``stability of $y$'') the quantities
$h_t(y)$ have the physical
interpretation of the ``density of free energy of the $y$--th Gibbs state
at the vertical level $t_{\nu}$'' .
\remark{Note}
We are concentrated, in this paper, in the investigation of a phase picture
for a {\it fixed\/} hamiltonian. The investigation of phase diagrams of
particular
models (notice that there are in principle infinite parameters
in the models like (3) above!) should be based on the study of
the mapping
$$ \ \ \{ \ \text{hamiltonian}
\longmapsto \text{the ground states of }\{h_t(y)\} \ \} \tag 1.2$$
using theorems from the differential calculus of (infinitely) many
variables (like the
implicit function theorem).
This may require a suitable technical modification of the definition
of the contour functional $F$ and the quantity
$h_t(y) $
\footnote{Such a modification could act on the nonground values
of
$y$ only;
the ground values of $h_t(y)$ have nontrivial
physical interpretation and there can be no arbitrariness
in their definition!}
to obtain as nice differentiability (even local analyticity)
properties of (1.2) as
possible.
\endremark
\definition{Acknowledgements} The second author (M.Z.)
thanks the Erwin Schr\"odinger Institute for 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''
(``stratified'') hamiltonian on $X$:
The hamiltonian will be a finite
range one (in what follows we consider a suitable norm on
$\zv$ e.g. the $l_{\infty}$ one)
$$
H_{\Lambda }(x_{\Lambda }|x_{\Lambda^c })=
\sum
\Sb A\cap\Lambda \ne\emptyset\\
\diam A \leq r
\endSb
\Phi _A(x_A)
\tag 2.2
$$
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)
{\it Gibbs states\/}
of the model, more precisely of the probabilities
\footnote{The probabilities
$ P_{\Lambda} ^{x_{\Lambda ^c}} (\cdot)$ are called
{\it finite volume Gibbs states\/} under boundary condition
$x_{\Lambda ^c}$. } which are
given in finite volumes
$\Lambda $ by formulas
$$
P_{\Lambda}^{x_{\Lambda ^c}} (x_\Lambda )=
Z(\Lambda ,x_{\Lambda ^c})^{-1}
\exp(-\frac{1}{T}H_{\Lambda}(x_{\Lambda }|x_{\Lambda ^c}))
\tag 2.3
$$
where $T$ is the ``temperature'' and the partition function
$ Z(\Lambda ,x_{\Lambda ^c})$ is
$$
Z(\Lambda ,x_{\Lambda ^c})=
\sum_{x_{\Lambda }}
\exp(-\frac{1}{T}H_{\Lambda}(x_{\Lambda }|x_{\Lambda ^c})) .
\tag 2.4$$
Suitable {\it infinite volume limits\/} will be constructed
from these finite volume Gibbs states.
\subhead Notes
\endsubhead
{\bf 1.}
In fact, some other (more special than (2.4)) partition functions --namely
so called (strictly) diluted partition functions will be
important later
and the recurrent structure of the measures (2.3)
-- which is formulated by the DLR equations --
will not be used explicitly in our later approach.
More adequate for our later approach is the idea that
$\Lambda$ is some (large) volume which will be
{\it fixed\/} in the main part of our future considerations.
(Only at the very end of the paper
-- when proving and interpreting our Main Theorem, section 3.8 --
this ``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 always put $T=1$\ i.e\. we
include the term $\frac{1}{T}$ into the definition of
$\Phi _{A}$ and $H$. Thus the temperature
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 the 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, the relevant result on
the behaviour of the phase {\it diagram\/} is not even
formulated in our paper!
\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
$x_{A}\ne\{x_t=\bar x_{t}, t\in A\}$
we obtain a limit Gibbs state on $\zv_{+}$ under the
boundary condition $\bar x$ on $\zv_{-}=\zv\setminus\zv_+$.
\newline {\bf 4.}
In fact, sensible and nontrivial results (requiring the full strength
of all the forthcoming constructions) can be formulated
even for a fixed {\it finite\/} volume $\Lambda$
(imagine the cardinality $|\Lambda | = 10^{27}$!), with suitable
boundary conditions.
However, in this 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 our study from the case $\zv$ to the case of
periodic boundary conditions, we expect that only
minor parts of the text should be adapted or
replaced by another arguments (for example
the parts of the text using the lexicographic ordering of $\zv$).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 1. Stratified configurations, hamiltonians and states
\endhead
For any $u\in\zw$ consider the shifts
%(we identify $u\in\zw$ with $(u,0)\in\zv$)
$ 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 \newline
{\it horizontally invariant\/} if
$$
U(x)=x\quad \text{for each } u\in\zw =
\{(t_1,\dots,t_{\nu -1},0)\} \subset \zv\,.
$$
\df We denote by $\es\subset \ex$ the collection of
{\it all stratified configurations\/}.
\newline Analogously we define the notion of a {\it stratified
hamiltonian\/} $H=\{ \Phi _{A} \}$ and a
{\it stratified\/}\ (Gibbs) {\it measure\/} $\mu $
by requiring
$$
\{ U\Phi _{A}\}=\{ \Phi _{A}\}\,,\,\,\,U(\mu )=\mu
$$
for each $u\in\zw$.
\remark{ Notes} {\bf 1}.
These will be the ``local
ground states'' of our model.
Of course only {\it some\/} of these configurations 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. We will see below that a nonexistence
of a substantial energetic barrier between the
``true local ground states''
and the remaining elements of $\es$ would deteriorate the validity of
the Peierls condition. The best solution in such a situation
seems to be to
choose the reference family $\es$ as big as above and to
look for energetical barriers between $\es$ and
configurations which are {\it not\/} stratified.
\footnote{This is an interesting methodological point even for the
ordinary Pirogov -- Sinai theory. 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.}
\newline
{\bf 2.}
The framework when all local ground states of the model are
assumed to be stratified (analogously: translation invariant
in the ordinary Pirogov -- Sinai theory)
seems at 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
{\it blockspin transformation\/} 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 a {\it degeneracy\/} of a (local) ground state one usually means the
fact that {\it several\/} (local) ground states exist
for a given hamiltonian. Here, we are looking
for (local) ground states
among the elements of $\es$ \
i.e. for the configurations $y \in \es$ such that
$\sum_A (\Phi_A(x_A) -\Phi_A(y_A)) > 0 $ whenever $x$ differs from
$y$ on a finite set whose vertical size is ``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 unicity\/} region. Recall that in the region
of phase {\it unicity\/} other, well developed methods of study
(based essentially on the Dobrushin' s
unicity theorem and later
investigations of the ``complete analyticity'' properties
by \cite{DSA}) are available.
On the other hand, in the regions
where phase coexistence is expected 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
Given a configuration $x\in\ex$ say that a point
$t\in\zv$ is {\it stratified\/} point of $x$, more precisely {\it
$y$-stratified\/} of $x$ (where $y\in\es$) \ if
$$
x_{\tilde t}=y_{\tilde t}
$$
holds for each $\tilde t\in\zv$ such that $|\tilde t-t|\leq r$.
\remark{Note}This is an analogy of the notion of a correct point of the
ordinary Pirogov -- Sinai theory; $r$ is the range of
interactions. %We extend this notion to configurations
%defined only on some {\it subset\/} $\Lambda \subset \zv$
%by requiring also $\tilde t \in \Lambda$ in the above condition.
\endremark
Say that $\Lambda\subset \zv$ is a {\it standard volume\/}
if any $t,t' \in \Lambda ^{c}$ with the same last
coordinate $t_{\nu}=t_{\nu}'$ can be connected by a
``horizontal'' (keeping the last coordinate intact)
connected path in $\Lambda ^c$
, i.e\. if all the sets
$C_n=\{t\in\zv;\, t_\nu=n\}\setminus\Lambda$ are connected.
A configuration $x\in\ex$ will be called {\it
$y$-diluted\/}, for $y \in \es$, if there is some
$
\ee\subset\zv
$
with standard finite connected components such
that all points of $\ee^c$ are $y$-stratified.
This value $y$ will be
called the {\it external colour\/} of $x$. We will use the notation
$y=x^{\text{ext}}$.
\definition {Precontours} For any diluted $x$ denote by $B(x)$ the collection of
all its nonstratified (i.e\. stratified for {\it no\/}
$y\in\es$) points. If $C$ is a connected component of
$B(x)$ then the pair
$$
\Gamma =(C,x_C)
$$
will be called the {\it precontour\/} of $x$ and we will write
$$
C=\supp \Gamma\,.
$$ \enddefinition
The prefix pre- suggests that the notion of a precontour is a
provisional one. It will be replaced below by a more elaborate notion
of a contour, with more convenient properties:
\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\/} diluted
configuration $x$. The configuration $x$ is uniquely
determined by $\Cal D $ only in all horizontal levels
intersecting $\Cal D $\,, otherwise it will be given
by the context (typically by the boundary conditions outside of
the given finite volume, in which we will be normally working)
and it will be denoted by $x_{\Cal D }$.\enddefinition
Of course,
this is a slight abuse of notations but we will see later that
the quantities defined below as functions of $x_{\Cal D }$ will
not depend, in fact, on this ambiguity in the choice of $x_{\Cal D }$.
We will also use the notion of an admissible system of precontours
$\Cal D $ in a given (standard or nonstandard) {\it finite\/} volume $\Lambda$.
In such a case, we will assume that a configuration $x =x_{\Cal D}^{\Lambda}$
can be defined in the whole lattice $\zv$
%a {\it neighbourhood of
%$\partial \Lambda^c \cup \Lambda $\/}
%Lambda^c $ denotes the collection of all points
%of $\Lambda^c$ having the distance at most $r$
%from $\Lambda$)
having the following
properties : i) $\Cal D$
is the collection of all precontours of $x =x_{\Cal D}^{\Lambda}$
and ii) all the points of $\Lambda^c$ are stratified points of
the configuration
$x =x_{\Cal D}^{\Lambda}$.
%\definition{Definition}
%A finite volume $\Lambda$ will be called a {\it standard\/} one
%if all the points of $\Lambda^c$ can be connected to infinity
%by a connected path in $\Lambda^c$ which is ``vertically''
%constant (keeping the last coordinate intact).
%\enddefinition
% \remark {Note}
% 1) In other words, all the components of any
% ``vertically constant''
% section of $\Lambda$ are simply connected
% if (and only if) $\Lambda$ is a standard volume.
%\newline
% 2) All volumes $\Lambda$ considered below
% will be the standard ones, if not stated otherwise.
% \newline
% 3)
% For volumes $\Lambda$ such that
% the horizontal sections of $\Lambda$ are not necessarily simply connected
% it can happen that the configuration $x$ mentioned above
%can {\it not\/} be continued to the whole lattice $\zv$. Imagine a
%three dimensional
% ``bumerang '' $\Lambda$ \ having different ``colours'' (boundary conditions)
% $y$
%on its different corners, in the same horizontal level.
%Then it can happen that an admissible collection of precontours
%in $\Lambda$ is not an admissible collection
%in the whole lattice.\endremark
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 3. Contours
\endhead
In the ordinary Pirogov -- Sinai theory, where elements of
the ``reference family'' $\es$ are constant
configurations, one immediately realizes that precontours are
``crusted'' in the sense that the (only!) infinite component
of $(\supp\Gamma )^{c}$ -- 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 ``disconnected'' from the exterior of $\gb$.
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 again 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'' (see below)
separating
various types of `` ceilings '' (i.e. various horizontally
invariant configurations in our case) and the possible sense of
the very notion of
an exterior resp. interior of a precontour surely
deserves a clarification.
With such ``noncrusted'' objects, one has to be more careful
when defining suitable {\it hierarchy\/} between them; the usual
concepts from the ordinary Pirogov -- Sinai theory like the concept of
an ``external contour'' or the notion of a contour
``inside of the other one'' must be defined
more cautiously and precontours are {\it not} the suitable objects
to do that. Imagine various precontours like ``fingers'' or
``bumerangs'' intersecting ``interiors'' of other precontours having
the shape of Dobrushin's
``walls'' -- for example the case of two walls each having an
``appendix, a
finger''
touching the ``interior'' of the other wall -- to be assured that the
notion of an external
or internal contour requires a careful definition here.
We should warn the reader that there will be {\it no\/}
telescopic equations in our approach.
However, the quantity $A(\gb)$ constructed in section III.3
substitutes these telescopic equations, in some sense.
%($\Gamma
%_1$ and $\Gamma _2$ on the picture will be glued together
%into a single contour):
The definition of a wall suggested by Dobrushin
(precontours ``of the interface'' are
called ``walls''in [D], and correspondingly also in [HKZ] and [HZ])
resolves these
problems in the special situation of one interface, by considering
the projection of the situation appearing at the interface to
the sublattice $\zet^{\nu -1}$.
However, this construction can be hardly transferred
to the situations where two or more parallel ceilings appear.
Thus, the proper definition of a contour in our situation (where general
stratified phases appear and
where possibly {\it many} rigid interfaces
appear in the considered phases)
cannot follow literally the concept of
the above mentioned ``wall''.
We will choose another aproach, which does {\it not\/} use an
auxiliary transcription of the situation to the dimension $\nu - 1$:
The crucial notion in our approach
will be that of the ``exterior'' (in $\zv$!) of an admissible
system of precontours ; alternatively the complementary
notion of an interior i.e. the volume ``swallowed'' by the given admissible
system of precontours:
(Notice that the definition below still follows essentially the original
Dobrushin's approach.)
\df
Let $\Gammab = \{\Gamma _i\}$ be an admissible system of
precontours in $\zv$ (or in a given standard volume $\Lambda$).
We denote by $\ext \Gammab$ the collection of
all points of $(\supp \Gammab)^c $ which can be accessed
from infinity %resp. from the complement of $\Lambda$ if we work with
%an admissible system in $\Lambda$ )
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. We denote by
$$ V(\Gammab)=(\ext\Gammab)^c \,.
\tag 2.5
$$
\remark{Note}
This is just 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
(in fact, this exception
will be really relevant only for some explanatory notes
interpreting in some detail
the structure of the phases constructed by Main Theorem)
we could work mainly with
{\it finite standard volumes $\Lambda$\/}
and always with a boundary
condition
$y \in \es$ given on the boundary of the volume $\Lambda^c$\ ;\
i.e. we will work
with diluted configurations having a finite
number of precontours only.
\endremark
Now we come to the definition of contours:
\df
Say that an admissible subcollection $\Gammab'\subset\Gammab$
is {\it removable\/} from the admissible collection $\Gammab$
(we will also alternatively
say that $\Gammab'$ is {\it interior\/} in $\gb$-- though this
characterization has not such a selfexplanatory
meaning as in
the ordinary, translation invariant, Pirogov --
Sinai theory ) if
$$
V(\Gammab')\cap\supp(\Gammab\setminus\Gammab')=\emptyset .
\tag 2.6 $$
and moreover if
%any two points from the boundary of $V(\Gammab')$
%with the same last coordinate can be connected by a connected
%``horizontal'' path avoiding
$ \Gammab\setminus\Gammab'$ is again an admissible system.
\remark{Note}
Imagine that the subsystem $\gb'$ was ``replaced by its external
colour'' (induced by $\gb'$ inside $V(\gb')$). Removability of $\gb'$ means
just
the {\it possibility\/} of
such a replacement .
\endremark
\definition {Definition}
An admissible collection $\gb$ of precontours in a given volume $\Lambda$,
with $V(\gb) \subset \Lambda$ and
with {\it no\/} removable
subfamilies will be called a {\it contour in\/} $\Lambda$.
\enddefinition
\remark{Note}
%herefore, all the paths mentioned in the definition of a removable
%ubsystem $\gb'$ belong either to $V(\Gammab\setminus\Gammab')$
%r to $\ext(\Gammab\setminus\Gammab')$ -- depending on the last coordinate.
%quivalently, this condition can be expressed by requiring that
%here is a set $\Lambda$ whose all horizontal sections (by hyperplanes
%\{ t : t_{\nu} = \text {const}\ \}$ ) are simply connected such that
%\supp \gb' \subset \Lambda$.
Thus, contours are some ``minimal '' -- in the sense that
no subsystem can be removed from them --
collections of precontours
which exist as admissible systems in the given volume $\Lambda$.
If all the horizontal sections of $\Lambda$
are even simply connected (i.e. have exactly one
component) then any contour in $\Lambda$
is also a contour in the whole lattice $\zv$.
Otherwise, the property ``being a contour'' depends on the
volume $\Lambda$ and the boundary condition, too .
For example, imagine a precontour having a shape of
a ``bumerang'' with different colours on its corners which are
assumed to have the same vertical level. This can be a contour
in a suitable bigger ``bumerang'' $\Lambda$ -- which will be
typically the interior of some other,``exterior'' contour.
However, it is not the contour in the
whole lattice $\zv$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 4. Representation theorem
\endhead
\definition{Definition}
We introduce the structure of an {\it oriented graph\/} on the
collection of all contours more precisely the following
``hierarchy'' between
the contours: Write $\Gammab\rightarrow\Gammab'$ whenever
$\Gammab\cup\Gammab'$ is admissible,
$V(\Gammab')\cap\supp\Gammab\ne\emptyset$ however
$\Gammab$ is removable from $\Gammab\cup\Gammab'$.%any standard volume
%ontaining $V(\gb \cup \gb')$ .
\enddefinition
\remark{Note}
The notation $\Gammab\rightarrow\Gammab'$ will be frequently
used, in an analogous sense, also for general {\it admissible
systems\/} (not only for contours) $\Gammab$ and $\Gammab'$.
\endremark
To say that $\Gammab$ is a contour of a
{\it given configuration\/} will not be so straightforward as
for the precontours (and as in the ordinary translation
invariant situations of the Pirogov -- Sinai theory).
However, the following representation theorem is still
valid:
\subhead Theorem 1
\endsubhead
Any diluted configuration $x\in\ex$ having a finite $B(x)$
is {\it uniquely \/} represented by a graph
on some finite subcollection of the collection of all contours of the
model; such a graph is always a ``forest'' of trees.
By a tree we mean an oriented
connected graph {\it without loops\/} i.e. without cycles
of bonds (cycles of ``arrows'') of the type $\gb_1 \rightarrow \gb_2
\rightarrow \dots \gb_n \rightarrow \gb_1 $. %The ``arrows''
%$(\Gammab,\Gammab')$ of the graph correspond to
%the situations where $\Gammab \Gammab'$.
The external contours
of the forest (namely those contours which are
not the starting points of some arrows
of the graph) have mutually disjoint volumes, and more generally
any subsystem
of the forest constructed by the rule ``if the end of the arrow is
removed then the beginning of the arrow is removed, too''
is an admissible system in $\zv$.
The mapping \{ configuration $\to$ forest of trees \} is
one to one;
in particular any forest corresponds to some
configuration of the original model.
An
analogous statement is also true for any diluted
configuration in any finite standard volume.
\footnote{Once again: any $x$ is uniquely
determined by the family of its contours.
This family is a ``forest of trees'' in the relation $\to$ .
Conversely, any such ``forest of trees of contours'' is an admissible
system i.e. it
determines (uniquely) some configuration.}
\remark{Notes}
0) The statement that {\it any\/} forest corresponds to some
configuration of the original model will not be used
in the following. What will be used only is the property that
the considered collection of forests is {\it horizontally translation invariant\/}.
\newline 1)\ %y a forest we mean more precisely
%a disjoint union of connected oriented graphs
%without loops which have no arrows between various connected
%components.
The generalization of the above result to general diluted configurations in
infinite volumes with infinite
$B(x)$ will not be considered here.
When formulating the properties of typical
configurations of infinite volume Gibbs states
(constructed as the consequence of our Main Theorem
at the very end of the paper), this could be done
by saying that any such configuration can be interpreted as
an {\it infinite\/} forest of {\it finite\/} trees (connected graphs of contours without loops).
However, one really needs
such formulations only for some explanatory notes commenting in more detail
the structure
of infinite volume Gibbs states constructed by our Main Theorem.
Otherwise, there will be no need for the consideration
of configurations in infinite
volumes in the rest of the paper.
\newline
2) To reconstruct the configuration $x$ {\it uniquely\/} from
the forest, we need to know also the {\it external\/} colour
$x^{\ext}$. Namely, the external colour of the forest
can be only partially recovered
from the contours of the forest. However, its value
elsewhere will be usually given by the context (by the
boundary conditions outside
of the finite volume, in which we will be actually working).
\endremark
\df \newline
Removable contours of the forest will be called the {\it
internal contours\/} of $x$.
A contour which can appear as a single remaining contour
after some succession of removals will be called the {\it
external contour\/} of $x$.
By a {\it removal} of \ $\Gammab'$\ from $\Gammab $ we mean the
replacement of $x_{\Gammab}$ by $x_{\Gammab\setminus \Gammab'}$.
\subhead {Proof of Theorem 1}
\endsubhead
It is based on the following
\subhead {Lemma 1}
\endsubhead
If $\Gammab $ and $\Gammab' $ are two different internal
contours of an admissible system $\de$ then
$$
V(\Gammab)\cap V(\Gammab')=\emptyset\,.
\tag 2.7
$$
\remark{Note}
By an internal contour of \,$\de$ we mean here (we have not yet
proven the theorem!) any {\it minimal possible\/} removable
subsystem of $\de$.
\endremark
\subhead Proof of Lemma 1
\endsubhead
Denote by
$$
C= V(\Gammab)\cap V(\Gammab')\,.
$$
We will show that the assumption $C\ne\emptyset$ would lead
to the removability of the system of all precontours of
$\Gammab$ contained in $C$
and this would be in contradiction with the fact that
$\Gammab $ is contour.
Denote by $y$ resp\. $y'$ the external colours $\Gammab$
resp\. $\Gammab'$: these configurations are defined
locally for any nonempty slice
$$
C_{\Gammab }^n=V(\Gammab )\cap\zv_n
$$
resp\. $C_{\Gammab'}^n$; here we denote by
$\zv_n=\{t\in\zv;t_{\nu}=n\}$.
For any $n\in\en$ such that
$$
C^n=C^n_{\Gammab }\cap C^n_{\Gammab'}\ne\emptyset
$$
define $y^*$ as the external colour of $C^n$.
Notice that all the points of $\partial C^c$ are stratified
(because they belong either to $V(\Gammab )^c$ or
$V(\Gammab')^c$) and thus we have either
$
y^*=y\quad \text{or}\quad y^*=y'
$
for any level $n$ such that $C^n\ne\emptyset$.
Notice also that all points of $\partial C^c$ can be accessed
``from the infinity'' by a connected horizontal path
belonging to $C^c$.
(We will not prove here this obvious fact, saying that the intersection
of two simply connected sets is again simply connected.)
Now there are two possible situations:
(1) If there is some $n\in\zet$ such that
$$
y_n^*=y_n'\ne y_n
$$
then we have the obvious relation (look at the level $n\,$!)
$$
\supp \Gammab \setminus C\ne \emptyset
$$
and therefore the admissible system $\Gammab^*$ of all
precontours of $\de$ contained in $C$ must
{\it not\/} contain all precontours of $\Gammab$.
However $\Gammab^*$ is removable (just replace $\Gammab^*$
by its external colour $y^*$ in $C$!) which is
a contradiction with the fact that $\Gammab $ is a contour.
(2) If $y_n=y'_n$ for all $n$ (where both colours are uniquely defined)
then
it is easy to see also that $y^*=y=y'$ on $\partial C^c$. Then we follow
an analogous argument as in (1): The condition $C\ne\emptyset$
would mean that the collection \ $\gb^*$
is also
a contour -- and this is not a contradiction (with the fact that
both $\gb$ and $\gb'$ are contours i.e. with no removable subsystem
\ $\gb^*$) only if $C = V(\gb) = V(\gb')$.
However, then e.g. $ \gb \setminus \gb^* $ must be empty because
otherwise the intersection of $\gb \setminus \gb^* $ and $V(\gb')$
would be nonempty and $\gb'$ would not be an interior
contour of the system.
\subhead Proof of Theorem 1
\endsubhead
We proved in Lemma 1 that interior contours of $x$ are
uniquely defined, with mutually disjoint volumes
$V(\Gammab )$. We can remove all these internal contours
(in an arbitrary order!) thus obtaining some new
configuration $\bar x$. Then we determine all the internal
contours of $\bar x$; after removing them from $\bar x$ we
obtain another configuration $\bar{\bar x}$ etc.
%Notice that any interior contour $\Gammab$ can intersect $V(\Gammab')$
%(of some internal contour $\gb'$ of $\bar x$ for {\it at most one}
% $\Gammab'$ only.
At the final step some collection of contours which are
both internal and external remains; their removal has as
its result the stratified configuration $x_{\ext}$.
The bonds of the forest representing $x$ are all the pairs
of the type $\Gammab \rightarrow \Gammab'$ where $\Gammab$
is an internal contour of some intermediate configuration
$(x,\bar x, \bar{\bar x},\dots)$ and $\gb'$ appears as an
internal contour after successive removal of all
$\gb$ such
that $\supp \gb \cap V(\gb')\ne \emptyset$\/.
%It is clear from this construction that a forest of
%trees is obtained
%(namely the bonds $\Gammab \rightarrow \Gammab'$ are
%constructed {\it after\/} the removal of $\gb$ and
%{\it before\/} the removal of $\gb$'),
%and a simple inductive argument can be used
%to prove the unicity of the graph thus constructed.
% Notice that in addition to the the bonds $\Gammab \rightarrow \Gammab'$
% {\it after\/} the removal of $\gb$ and
% {\it before\/} the removal of $\gb$'; other bonds of this type
%can appear in the tree constructed above, but not between the
%nearest neighbors.
\remark{Notes}\newline
1. Of course one can easily construct examples of ``loops of contours''
$\gb_1,\dots, \gb_{n+1}=\gb_1$ such that $\gb_i
\rightarrow \gb_{i+1}$ for each $i=1,\dots,n$ but {\it no\/}
such cycles can appear in the representation by Theorem 1.
They form a {\it single contour } there!
\newline
2. Analogously, one can formulate an analogous representation
theorem for configurations in finite standard volumes.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 5. Connectivity of contours
\endhead
\subhead Theorem 2
\endsubhead
Contours are ``halfconnected'' in the sense that
$$
|\supp\gb|\geq {64\over 81} \con \gb
\tag 2.8 $$
where $\con\gb$ denotes the minimal possible
cardinality of a connected set containing $\supp\gb$.(Compare also
Theorem 2' in Section II.6.)
\remark{Note}
The notion of $\con \gb$ will be {\it modified\/} in later
sections,
its new value (denoted by $\conn \gb$) being actually smaller than $\con \gb$
(however without having
a simplifying effect on the proof below.)
\endremark
\subhead Proof
\endsubhead
We have to understand the structure of contours of our model -- which are
defined as some complicated ``conglomerates'' of (connected!)
precontours. We present here only an outline of the
proof leaving out the ``topological'' details.
\df
Let $\{\Gamma _i\}$ be an (unadmissible) family of precontours.
Say that the two points $t,s \in \partial (\cup_i
\supp \Gamma_i)^c$ with the same height $t_{\nu}=s_{\nu}$ are in
a conflict in $\{\Gamma _i\}$ if
their colours (induced by neighboring precontours) are
different and at the same time $t$ and $s$ {\it can\/} be connected
by some connected horizontal (keeping $t_{\nu}$ constant) path
not intersecting
$\cup_i \supp \Gamma_i$. %(and remaining in the given volume ).
Say that $\Gamma $ {\it cures} the conflict between $t,s$
if $t,s$ are no more in conflict in $\{\Gamma _i\}$ \&
$\{\Gamma \}$.
\subhead Lemma
\endsubhead
Let $\Gamma_1,\Gamma_2,\dots\Gamma_n$ be a sequence of
precontours such that each $\Gamma _k$ cures some conflict
in $\{\Gamma_1,\Gamma_2,\dots\Gamma_{k-1}\}$.
Then $\{\Gamma_1,\Gamma_2,\dots\Gamma_{n}\}$
is ${8\over 9}$connected.
\subhead Proof of Lemma
\endsubhead We will
construct for each $\{\Gamma_1,\Gamma_2,\dots\Gamma_k\}$
a connected set $M_k\supset\cup_{i=1}^k \supp \Gamma _i$
by the following inductive procedure:
Connect each $\Gamma _k$ to the (yet constructed)
``connected conglomerate''
$M_{k-1}\supset\cup_{i=1}^{k-1}\supp \Gamma _i$
in the following way:
there must be some horizontal section
$$
C^n_k=\supp\Gamma _k\cap\zv_n
$$
such that in some internal component of $(C^n_k)^c$ (taken in the
lattice $\zv_n$), some
points of
$\cup_{i=1}^{k-1} \Gamma _i$
can be found.
(Otherwise $\Gamma_k $ would cure no conflict.)
Now take $M_k$ as the union of $M_{k-1}$ and $\supp \Gamma _k$
and of some shortest path connecting $C^n_k$ and
$M_{k-1}$. Clearly, the lenght of such a shortest path is no
more than $\frac{1}{8}$ of the cardinality of $C^n$.
\df
Say that an admissible collection
$\{\Gamma_1,\Gamma_2,\dots\Gamma_{n}\}$
cures $\Gamma _1$ if each $\Gamma _k$,
$k\leq n$ cures some conflict
$\{\Gamma_1,\Gamma_2,\dots\Gamma_{k-1}\}$. \newline
%Now, we will need a straightforward generalization of the notion of a contour
%(which will be used {\it only\/} in this proof ).
%
%\df
%Define the concepts of an admissible collection and of
%a contour in an analogous way as before for a configuration in any
%{\it subset\/} $\Lambda \subset\zv$ and for any boundary
%condition which is {\it locally\/} stratified on $\partial (\Lambda ^c)$:
(Recall that an admissible family of precontours in $\Lambda$ is
a family of precontours whose
``outside colours'' are
not in mutual conflict.)
%Define now the famillies of conflicted precontours
%in $\Lambda $ as before --allowing the constructed connected
%paths to enter also $\partial (\Lambda ^c)$.
\subhead Proof of Theorem 2
\endsubhead
Take some precontour $\Gamma _1\in\gb$ and cure it
successively to obtain some ${8\over 9}$ connected subcontour
$\tilde \gb_1 $ of $\gb$. Denote by
$\gb_1$ some maximal ${8\over 9}$ connected supersystem of
$\tilde \gb_1$ in $\gb$.
Asssume that $\gb^{1} = \gb \setminus \gb_1$
is nonempty.
(Otherwise, the proof of the theorem would be complete.)
Take some $\Gamma _2 \in \gb^{1}$ and cure its conflicts
analogously as $\gb_1$ was constructed, and so on.(Notice that the
connecting paths starting from the precontours curing some
conflicts in $\Gamma _2$ etc. do not touch $\gb_1$ yet!)
%(Notice that both $\gb_1$ and
%$\gb\setminus\gb_1$ are {\it non\/}removable from $\gb$.)
%Analogously, in $\Lambda =\zv\setminus\supp\gb_1$
%under the boundary condition
%$\gb_1$ on $\partial \Lambda ^c$, take some precontour
%$\Gamma _2$ and cure it to obtain a
% {8\over 9}$ connected contour in $\Lambda $
%etc\.
Thus we obtain some decomposition
$$
\gb=\bigcup\limits_{i=1}^{N}\gb_i
\tag 2.9 $$
where $\gb_i$ are ${ 8\over 9}$ connected contours such that
$ \cup_{i=k+1}^{N}\gb_i
$
is nonremovable from $\cup_{i=k}^{N}\gb_i$.
%(in $\zv\setminus \bigcup\limits_{i=1}^{k-1}\supp\gb_i$).
Take the admissible collection
$\cup_{i=1}^{N-1}\gb_i$ \ i.e\.\ $\gb\setminus\gb_N$ \
and denote by $\de_1, \dots, \de_m$ the internal contours of
this truncated system. Notice that {\it all\/} $V(\de_i)$
must be intersected by $\gb_N$ (those nonintersected by
$\gb_N$ would be {\it removable\/} from $\gb$!)
and therefore $\gb_N$ looks like a ``bumerang'' such that
$\Cal D_i$ are like some ``rings'' entwining it.
%(Write a picture.)
%(contours having the shape of a bumerang
% entering interiors of {\it several}
%$\de_i$ can appear only here in this more general provisional setting,
%not in the normal setting when working in the whole volume
%$\zv$ )
Clearly, we can connect $\gb_N$ to any $\de_i$ ``at the
expense of $\de_i$'' by some shortest path from
$\gb_N\cap V(\de_i)$ having a lenght at most $\frac{1}{8}$
of the cardinality of $\de_i$.
For any $\de_i$ only {\it one\/} path will be constructed.
This procedure of ``making connections above'' can be
replaced also for the bigger ``bumerang''
$\gb_N\cup\cup_{i}\de_i$
``at the expense of the internal contours''
of the system
$
\gb\setminus(\gb_N \cup\cup_{i}\de_i)
$
etc\.
Assuming yet that we have submerged $\supp\gb_i$ into some
connected set $C_i^*$ such that
$$
\card C_i^*\leq (1+\frac{1}{8}) \card\supp\gb_i
\tag 2.10 $$
we can submerge $\supp \gb $ into some connected set $C^{**}$
such that
$C^{**}=C^* \cup P$,
where
$P$ is the support of connected paths constructed above
and
$$\card C^{**}\leq (1+\frac{1}{8})\card C^*\leq
\frac{81}{64}\card\supp\gb .$$
\head 6. Supercontours
\endhead
The contours defined so far would still have some inconvenient
features later. Remember that they are ``not crusted'' -- in the sense
that the internal contours $\gb$ intersecting some other $V(\gb')$ need
not to satisfy the condition $ V(\gb) \subset V(\gb')$.
\footnote{Such a
property was valid, it seems to us, in all the previous applications
of the Pirogov -- Sinai theory.}
This is a fundamental obstacle -- which apparently can {\it not\/} be
remedied by some ``better'' definition of a contour.
However, even the fact that the external contour $\gb'$ can be
much ``smaller '' than $\gb$ would be rather
inconvenient in our following considerations.
(This will be so for pure technical reasons ;
see part III, Theorem 5 and also Theorem 7.)
The latter inconveniency can be, however, remedied;
one can
redefine contours such that $\gb'$ is always ``substantially
bigger'' than $\gb$
if $\ \gb \rightarrow \gb' $ : Below, we will ``glue together
some contours'' to achieve this property, keeping still the validity of
Theorem 2 (with a slightly smaller constant) for the newly defined
conglomerates of contours. These conglomerates will be called supercontours
and below (in Part III ) we will work exclusively with them (instead of
working with contours).
\definition {Definition} For any system of contours $\Cal D$
introduce its ``diameter''
$$ \diam \Cal D = \max_{\gb \in \Cal D} \diam \gb . \tag 2.11 $$
Consider some (total) ordering $\prec$ on the
set of all systems of contours satisfying the following
requirements : 1) if $\diam \Cal D < \diam \Cal D' $ then
$\Cal D \prec \Cal D'$ ; 2) if $\Cal D \prec \Cal D'$ then
$\diam \Cal D \leq \diam \Cal D' $ ; 3) if $\Cal D$ is a subsystem
of $\Cal D'$ then $\Cal D \prec \Cal D'$ ;
4) if $\Cal D = \Cal D' + t $ where $ t \prec 0 $ in the lexicographic order
on $\zv$ then $\Cal D \prec \Cal D'$.
\enddefinition
\remark {Notes}
The particular choice of a norm on $\zv$ is not relevant here.
However,
in view of the further usage of the notion of a diameter in part
III let us make the agreement that from now on the $l_{\infty}$ norm
$|t| = \max\{t_i\}$ will be used everywhere in what follows.
The lexicographic order on $\zv$ is assumed to be
{\it fixed throughout the paper\/}.
The requirements 1), 2), 3) and 4) clearly define a
{\it partial} order and we simply extend it to some total ordering
of the family of all systems of contours.
\definition{Definition} Say that a contour of the ``forest'' of Theorem 1
has an index $n > 0$ if it is an internal contour of the
forest remaining after i) the
removal of
all the {\it internal\/} contours of the forest (these contours will be said
to have the index $1$) and then after ii) the successive removal
of all the
contours having the index $ 2, 3,... , n-1 $.
For any contour $\gb'$ having the index 2 find the {\it biggest \/}
(in $\prec$ ) contour $\gb$ such that
$\gb \rightarrow \gb'$. If $\gb $ is bigger than $\gb'$ or
the cardinality of $\supp\gb$ is bigger than the cardinality
of $\supp \gb'$
connect the contour $\gb$, by a shortest connected path, to $\gb'$.
Moreover, if the cardinality of the support of the
conglomerate thus formed by $\gb$
and $\gb'$ (and their connecting path )
is still not at least twice bigger than the cardinality
of $\supp \gb'$
for any other interior $\gb''\rightarrow \gb$ repeat the procedure
once again with $\gb''$ instead of $\gb$.
Repeat the same process with the collections of all remaining
interior contours (kept intact after the glueing procedures
above)
$\gb \rightarrow \gb' $ which are bigger (in the sense above)
than some contour
$\gb'$ having the index 3 . Then repeat the same glueing procedure
with contours having the index 4 etc.
Finally remove all the remaining interior contours
(not used in the glueing procedures above for $ n =2, 3, \dots $)
and repeate the whole above contruction again and again.
{\it Supercontours\/} of the original forest are then defined
as the connected (by paths constructed above) conglomerates
of contours of the original forest. The notion of a supercontour
includes also all the contours left intact by the
above procedure.
We will use the short name
supercontour for a supercontour of {\it some\/} forest. Notice
that the relation $\rightarrow$ between supercontours is defined
in a natural way because glueing cancels the arrow $\gb \rightarrow \gb'$
but does {\it not\/} effect the remaining arrows of the forest.
%(Imagine that the connection
%is done ``at the expense of $\gb'$ ''; notice that any
% contour of the forest is used at most once in such a construction. )
\enddefinition
\subhead Theorem 2'
\endsubhead
The new forest obtained by the construction above is uniquely
determined by the old forest. Any forest of supercontours
whose external supercontours form an admissible system
can appear as the result of the construction above. The relation
$\gb \rightarrow \gb'$ is a {\it subset\/} of
the relation $ \gb \prec \gb'$ for all pairs of supercontours of
the given model. Moreover, the
cardinality of $\supp \gb'$ is always {\it at least twice bigger
\/} than the cardinality of $\supp \gb$.
Supercontours satisfy again the
statement of Theorem 2, with the constant $64/81$
replaced by some smaller constant, e.g.
$$ |\supp \gb| \geq 1/2 \ \con \gb. \tag 2.8' $$
%%%%%%%%%%%%%%%%
\remark{Note}
These properties will have no special importance in the rest of part II
-- but they will be quite convenient later in Theorem 5
and also in Theorem 7.
\endremark
The {\it proof\/} is quite straightforward. To see
that Theorem 2 remains valid notice that the connecting path
constructed in any step of the construction of
a supercontour is done ``at the
expense'' of the external contour $\gb'$; notice that any
contour $\gb'$ of the original forest is
used at most once in such a construction, and to
construct a connected path ``inside'' $\gb'$, one does not need
more than $1/2$\ $ \con \gb'$ points (and ${{81 \cdot 2}\over {64\cdot 3}
} < \frac {1}{2}$).
In the rest of the paper, we will work exclusively with {\it
supercontours\/}
(of a given admissible system of precontours).
We will {\it omit\/} the prefix ``super'' in the following.
(However, this will be important only later, in part III.)%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 7. Expression of the hamiltonian
\endhead
For any stratified configuration $y\in\es$ define its
``density of energy'' at $t\in \zv $ \ :
$$
e_t(y)=\sum_{A\ni t}\Phi _A(y_A) |A|^{-1} \,.
\tag 2.12 $$
Given a contour $\gb$ one would like to define also a quantity
having the meaning of the ``contour energy''. One could think
for example about the
``energy excess'' of $H(x_{\gb})$, where $x_{\gb}$ denotes
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 -- when the density of energy inside $\gb$ is
the {\it same\/} as outside of $\gb$, at {\it any} horizontal level.
Otherwise, it will be necessary (to keep the interpretation
of a contour energy as a quantity which is ``localized on $\supp \gb$ '')
to replace the quantity
$H(x_{\gb})-H(x_{\ext})$ (which will be, of course, also very important later --
see (2.19)) by the following, perhaps too formally defined at first
sight, quantity:
first extend the notation $e_t(x)$ for any (even {\it nonstratified\/}
in $t$)
$x$ by putting
$$
e_t(x)=e_t(\hat x)\,; \quad
t=(t_1,t_2,\dots,t_{\nu})
\tag 2.13 $$
where $\hat x$ is the stratified continuation of the
vertical section
$\{x_{(t_1,\dots,t_{\nu-1},(\cdot))} \}$.
Now define the configuration $x=x_{\gb}^{\text{best}}$
{\it minimizing\/} the sum (notice that the terms of this
infinite sum are fixed outside $\supp \gb$!)
$$
\sum_{t\in\zv}e_t(x)
\tag 2.14 $$
under the condition that $x=x_{\gb}$ on $(\supp \gb)^c$
and also on the set
$$\partial\supp\gb=\{t,\dist(t,(\supp\gb)^c)\leq r\}\,.
$$
Put
\footnote{Of course this is again only a formal expression -- but obviously
the terms in the sums on the right hand side of the equation for $E(\gb)$
can be reorganized such that
a sum with only a finite number of nonzero terms is
obtained.}
$$
E (\gb)=H(x_{\gb})-\sum_{t\in\zv}
e_t(x_{\gb}^{\text{best}}) \ \ \ \text{where} \
\ \ H(x) = \sum_{A \subset \zv}
\Phi_{A}(x_A).
\tag 2.15
$$
Then we have the following expression of the hamiltonian ($\Lambda$
will be always a finite set in the sequel) :
\subhead Theorem 3
\endsubhead
Let $x$ be a diluted configuration in $\Lambda$. %such that $\supp
%\gb\subset \Lambda $ for any contour of $x$ and such that
%$x=y$ on $\partial \Lambda^c $ where $y\in \es$ locally.
Then
$$
H(x_\Lambda |x_{\Lambda^c})=\sum_{t\in\Lambda }
e_t(x^{\text{best}}_{\Cal D})+
\sum_{\gb \in \Cal D} E(\gb)
\tag 2.16 $$
where $\Cal D$ denotes the system of all contours of $x$
and $ x^{\text{best}}_{\Cal D}$ is defined as
above (with $\Cal D$ instead of $\gb$), by (2.14).
\subhead Proof
\endsubhead
Immediate, if we notice that $E(\gb)$ is a local quantity
(depending on $\gb$ only) and also an additive one:
$$E(\gb\cup\gb')=E(\gb)+E(\gb') .$$
Notice that whenever $\gb$ is the unique contour of $x$ then (2.16) follows
directly from (2.15).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 8. The Peierls condition. The abstract Pirogov -- Sinai model
\endhead
In the following we will assume that for any set $G$ and
any stratified configuration $y\in\es$ the
following (Peierls type) inequality
holds with a sufficiently large constant $\tau>0$:
$$
\sum
\Sb \gb:\ \supp \gb=G \ \& \ x_{\gb}^{\ext}=y\endSb
\exp(-E(\gb)) \leq \exp(-\tau|G|)
\tag 2.17
$$
We recall that we include the inverse temperature into the
hamiltonian and therefore $\tau$ is of the order of the
inverse temperature.
\remark{Notes}
1.In practice, one usually establishes (2.17) through the inequalities
$$ E(\gb) \geq \tau^* |G| \tag 2.17* $$
with suitable $\tau^* \geq \tau$.
2. In the following we will work {\it exclusively with the
expression\/} (2.16). It is generally advisable
to develop the Pirogov -- Sinai theory in an abstract setting (2.16)
-- with the Peierls condition (2.17) established.
The reformulation of the original model to the language
(2.16) can be considered as a suitable
``preparation'' of the given ``physical model'' -- and the Pirogov -- Sinai
theory actually only {\it starts} with the setting (2.16)\ \&\ (2.17).
3. Such a
preparation of the model (i.e. a
conversion to the form (2.16) by a suitable definition
of the notion of a contour) is often not unique
i.e. it is not ``naturally determined''
by the given model . One
can adapt it in various ways for various concrete
situations -- by modifying the concept of a
``stratified point'', for example, or even by considering
contours as objects different from those we constructed here.
Remember, for example, that in the ordinary theory of
the low temperature Ising model, contours are commonly defined as
selfavoiding {\it paths} in
the dual lattice.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 9. Diluted and strictly diluted partition functions
\endhead
In addition to the volume $V(\gb)$ there will be another
(actually more important in the following) modification of this notion,
denoted by $\vv$.
The details of its definition will be important
only later.
See also (3.19) in part III;
$\vv$ will be defined here as a suitable ``cone over $V(\gb)$'',
more precisely as follows:
\footnote{ The ``protecting zone''
$\vv$ over $V(\gb)$ constructed below is apparently unnecessarily big.
This will make no harm, however.
In later sections of part III, we will be more strict
when tackling analogous difficulties as in the definition
(3.19). The logarithmic
height
of the ``caps'' of $\vv$ over $V(\gb)$
would be quite sufficient, in fact.}
\definition{Conoidal sets}
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 $t_{\nu}=m$)
the whole ``cone'' $\{t \in \zv : \dist(t,B) \leq \frac{1}{2} \dist(t,
\partial B)\}$ where $\partial B$ denotes the boundary of $B$ taken
in $\zv_m$. If $\gb$ is a contour or an admissible system we define
$\vv$ as the union of $\supp \gb$ and the smallest conoid
containing $V(\gb) \setminus \supp \gb$. Equivalently, this is the union
of $V(\gb)$ and of the smallest conoid containing all the ``upper
and lower ceilings of $\gb$''
i.e. the flat parts of the boundary of $V(\gb)$ which are outside of
$\supp \gb$. \enddefinition
We denote, from now on, by symbols
$ Z^y(\Lambda )$ resp. $ Z^y_{\updownarrow}(\Lambda)$
the partition functions
$$ Z^y(\Lambda )=\sum\exp(-H(x_\Lambda |y_{\Lambda ^c}))
\tag 2.18 $$
where the sum is over all diluted configurations whose all
contours satisfy the condition
$$
\dist(V(\gb),\Lambda ^c)\geq 2\, \tag 2.18'
$$
resp. analogously for
$ Z^y_{\updownarrow}(\Lambda)$,
$$ \dist(\vv),\Lambda ^c)\geq 2\,. \tag 2.18'' $$
Of course, for conoidal $\Lambda$ the both
partition functions
(2.18') and (2.18'') are the {\it same\/}.
The rest of this paper (and the essence of the Pirogov -- Sinai theory in its
presented version) consists of the effort to {\it expand\/}
the considered diluted partition functions (3.18) (more precisely (3.18''))
as far as it is possible or reasonable --
in order to deduce some useful corollaries from these
expansions.
The attempt to expand partition functions (2.18) can be based on
the older idea of a contour model \cite{PS} (or of a metastable
contour model \cite{Z}). Though this notion in fact
{\it disappeared} from the presented version of Pirogov --
Sinai theory
-- instead of speaking about suitable ``metastable contour models''
we will work, in fact, only with {\it expansions} of their partition
functions (i.e. the partition functions
of the metastable submodels of the given model) --
it is perhaps 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, more
technical constructions.
We will see later 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 a central importance!
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 10. 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 Gibbs
states whose almost all configurations are some ``local''
perturbations of the considered reference (stratified)
configuration. More precisely, they are ``$y$-diluted''.
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'' $F(\gb)$
(the ``work needed to install the given contour'')
which ``tests'' those contours whose appearance as of
external contours of the metastable model is allowed.
To get an idea of such a testing quantity let us start with
its ``zero temperature version'':
Put
$$
F_0(\gb)=H(x_\gb)-H(x_\gb^{\ext})=E(\gb)-A_0(\gb)
\tag 2.19
$$
where
$$
A_0(\gb) = \sum_{t\in V(\gb)}
(e_t(x_{\gb}^{\ext})-e_t(x_{\gb}^{\text{best}}))
\tag 2.20 $$
This quantity is just a first approximation to the more
relevant quantity given at this moment only {\it formally\/}
by
$$
F_{\text{formal}}(\gb)=
\log Z^y(\zv)-\log Z^{\gb}(\zv)
\tag 2.21
$$
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
$\gb$''.
Below we will define, by relations (3.21) \& (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 contour
used below; the term $A_0(\gb)$ typically satisfies
an estimate like
$$
A_0(\gb)\leq C|V(\gb)|
\tag 2.22 $$
with a constant $C$ which is sufficiently smaller than $\tau$ (imagine
the Ising model with a small external field) and therefore if
e.g\.
$$
C|V(\gb)|\leq \frac{\tau}{2}|\supp \gb|
$$
(this will hold for contours which are ``not too big'')
we have, from the Peierls condition (2.17*), the inequality
$$
F(\gb)\geq E(\gb)-C|V(\gb)|\geq
\frac{\tau}{2} |\supp \gb|\,.
\tag 2.23 $$
We see that $y=x_{\gb}^{\ext}$ is really a ``local ground
state'' because installing of a ``not too big'' contour
increases its energy.
Unfortunately, it is not trivial to define
quantities like $F_{\text{formal}}$ rigorously. While
such a task is solved rather straightforwardly in other
situations 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 ``ceilings''
(flat horizontal parts of boundaries of $V(\gb)$ which do
{\it not\/} belong to $\supp \gb$) causes
problems! These problems lead to the necessity
of considering of
suitable {\it expansions\/}, and this is the main subject of
the forthcoming part of the paper.
\head III. The Concept of a Mixed (Partially Expanded) Model. Recoloring.
\endhead
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
This, the main and the final part of the paper 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 notion of a ``metastable
model'' (see [Z]) can {\it not\/} be (apparently) defined here without the
language of expansions;
even the formulation of our Main Theorem uses this language.
Of course the very 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 but
mostly as an important but auxiliary tool while in our
formulation it is really the cornerstone of the theory.
Our basic expansion step (Theorem 5, Theorem 4 and the Lemma preceding it)
-- called recoloring
by us {\it incorporates\/} some of the usual cluster expansion ideology
(based on expansion by power series and on the use of equations
e.g. of Kirkwood -- Salsburg type)
into the very construction of the contour functional.
A consequence of our approach is that 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)
will not yet give the required
expansion of the model. It must be {\it repeated\/} (infinitely)
many times. 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 with suitable graphs (without cycles)
on $\zv$.
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. removing of $\gb$ from the model $\&$ adjusting
of the new cluster series such that the partition functions would not
change) of a single interior contour $\gb$;
in the context of a general
mixed model. The important concept of a recolorable contour
(or admissible subsystem) 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 translation invariant one if the original
mixed model satisfied this property.
Technically, sections 3 and 4 (and later section 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 system of contours in a general
mixed model.
Namely, to have more specific examples of recolorable systems we
introduce there
a related but
better controllable notion of a small resp. {\it extremally small\/}
system of contours (which is more useful than mere
notion of a recolorable system).
The message of the sections 5 to 7 is roughly speaking the following:
once there are some small contours in the mixed model then
there is still ``something left to recolor''
i.e there are still also some recolorable subsystems in the model.
The notion of an extremally small system is an
elaboration of the older idea of a ``small'' or ``stable'' contour
(\cite {Z}). Notice that small resp. extremally small systems
can contain the
``large'' (``not extremally small'') contours or admissible systems
as its
{\it internal\/} subsystems.
Theorem 6 is proved with the help of Theorem 7; the latter is
already some general statement about the ``connectivity constant''
of some special (``tight'') sets appearing in the
study of extremally small systems.
Only after finishing the sequence
of all the expansions (recolorings) organized by us
we will be able to say what the {\it metastable\/} model
is -- in Section 8. This will be the submodel
of the original abstract \ps model where only those configurations
containing {\it no\/} ``redundant''
(i.e. surviving in the ``fully expanded'' model: by the
fully expanded model we mean the final mixed model remaining at the moment
when the inductive procedure of its partial expansions was completed)
external contour resp. admissible system 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 fully expanded model under such a boundary condition
i.e. the metastable model corresponding to $y$ gives an appropriate
$y $-- th Gibbs state.
The fact that under ``stable''
boundary conditions, ``everything is recolorable''
(i. e. the complete expansion of the partition functions is obtained)
is the
core of the proof of the main theorem. Having proved the
preparatory Theorems 5,6, this is now almost
a tautology.
Our new method based on Theorem 5 and Theorem 6
replaces the previous coarser arguments
from \cite{Z} (which moreover cannot be used in these new situations).
However, even in the situation
of \cite{Z} our new method is simpler (at least conceptionally:
inequalities for the partition functions employed in \cite{Z}
are now systematically
replaced by corresponding expansions whenever possible) and more powerful.
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
often written, in the literature, in the following form:
$$ \log Z^y(\Lambda ) =\sum _{T \subset \Lambda} k_{T} \tag 3.0$$
where $ Z^y(\Lambda ) $ is
the considered (diluted) partition function
in volume $ \Lambda $ under a boundary condition
$y \in \es$ and \ $k_{T} = k^y_T, T \subset \zv$ are some local quantities
(indexed by ``connected clusters'' $T$ -- see below for more
details about this notion) which are ``quickly decaying''
e.g. like (this will be the form used below by us)
$$ | k_{T}| \leq \varepsilon^{\conn T}
\tag 3.1 $$
where $ \conn T $ is something like the ``cardinality of a
minimal connected
set containing the cluster $ T $''.
( See below
in (3.6) for
the definition
of a quantity $\conn T $ which will be used in our later considerations.)
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 says that the quantities $ k_{T} $
are in fact sums of quantities indexed by some
``clusters of sets (resp. of contours) $\{\gb_i\}$''
(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
-- which was the typical application of the cluster expansions in
most previous variants of the P. S. theory) here it
will be necessary to retain the more precise information because
these expansions {\it 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 slightly changed quantity
denoted by $ \Conn T $ which is defined in a more direct
way.
\remark{Note}
The value of $ \con T $ used in the definition
of a contour in part II is inconvenient here and cannot be reasonably
used in what follows. However, our new value denoted by $ \conn T $ will
be actually {\it smaller\/}
than the corresponding quantity of section II and this
will enable to transfer immediately the estimates of the type
(2.8),(2.8') which were established in part II (and later combined
with the Peierls condition) to our present context.
\endremark
In the definition of $\conn T$ (see Definition 2 below)
we will use the notion of an abstract {\it tree\/},
often also with a specified {\it root\/}:
\definition{Definition} An abstract {\it tree\/} % with a marked root
is defined as an equivalence class, with respect to the isomorphisms of graphs,
of unoriented graphs without cycles. (By a cycle of a graph $G$
we mean a collection of the
type $\{t_1,t_2\},\dots,\{t_n,t_1\}$ composed of bonds of $G$.)
If we want to specify also the {\it root\/} of such a tree
(i.e. mark one point of the graph)
then such an object can be defined also in a 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}
The identification of a cluster with a suitable tree
-- which will be given below --
suggests that the following idea of the summation
of cluster expansion series will be developed:
instead of estimating the number of various clusters with the same
length we rather employ here the idea of the summation over the
trees (based on the recursive summation over the outer
bonds of the tree).
It seems that this method gives good estimates.
Therefore, we are following this method here,
in spite of the fact that the treatment given below is maybe
too much general for the purposes of the forthcoming text.
Namely, a weaker version could be apparently also made which would be closer to
our later approach of section 7 -- which is
based on the notion of a tight set; see the proof of Theorem 7.
Nevertheless we keep the method of summation over trees here,
also as a suitable
reference for possible further applications
of the method (like the paper [COZ] which is under preparation).
\endremark
\definition{Definition 1}
By a {\it commensurate tree on $\zv$\/}
we mean the following object:
\roster
\item It is a pair $\Cal T = (G,\phi)$
consisting of {\it an abstract tree \/} $G$
%(a connected graph without loops; isomorphic graphs are considered as identical)
and a {\it mapping\/} $\phi$ of this abstract tree $G$ to $\zv$; the mapping
can be constructed, after fixing of some root of the given abstract tree,
also recursively (according to the recursive definition
of an abstract tree given above): the image of the newly added root
is specified at each stage of the construction.
The vertices of the
abstract tree $G$
are mapped (generally
not one to one) to some subset of $\zv$ which will be called the
{\it support\/} of the tree (denoted by $\supp \Cal T$).
Notice that possibly several
vertices of the
given abstract tree $G$ can be mapped to the same
$t\in\zv$. Then these vertices of the tree $\Cal T$ will be sometimes
denoted by symbols $t'$, $t''$,
$t'''$
\dots to distinguish them.
\item The {\it bonds\/} of the tree $\Cal T$
constructed in (1) are (unordered)
pairs of the 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 bonds of the type $\{\phi(a), \phi(b)\}$
(where
$\phi(a) \in \{t', t'', \dots\}$ and $\phi(b)\in\{s',s'' \dots\}$)
which are images under $\phi$ of the corresponding
bonds 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\/}
is meant here in the sense that if $\{A = \phi(a),
B =\phi(b)\}$ and
$\{A= \phi(a), C=\phi(c)\}$ are two bonds then
$$ {1 \over 2} \rho(A, B) \leq \rho(A, C) \leq 2 \rho(A, B)\, \,
\tag 3.2 $$
where the distance $\rho(A,B)$
between $A \in \{t', t'', \dots \} $ and $B\in \{s', s'',
\dots\}$ 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 number of its bonds {\it excluding\/}
all the bonds (``loops'') of the type
$\{t', t''\}$.
\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{Definition 2}
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 $\{t', t''\}$.
%The {\it root\/}of the tree will be chosen as some arbitrarily selected point of $T$.
We will denote such a 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 $\{t',t''\}$ 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 of the tree having the length {\it at most
\/} $2$.
Define the auxiliary quantity $\Conn T$ as the {\it length\/}
of the tree
$\Cal T(T)$. %(``Loops'' $\{t',t''\}$ are not counted!)
In the following, it will be more useful to have a
modified version of this quantity, denoted by $\conn T$ and defined as
follows:
$$
\conn T = \Conn T +[3 \nu\log_2\diam T].
\tag 3.4 $$
\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 contours) ``whose diameters are not smaller than their distance
to other objects 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 \ $\approx \log\diam T$ steps:
\endremark
\proclaim {Lemma 1}
Let $\rho(t, s) = d$. Then there is a commensurate {\it path\/}
starting in the loop $\{t, t\}$ and ending in the loop $\{s, s\}$
having the length at most equal to
$[\,3 \nu \ log_2 \ d\, ]$.
\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
$$ 1, 2, 4, \dots, 2^k, \dots, 4, 2, 1$$
which overcomes the distance $d= 3\cdot 2^k -2$. The length of this path is
$2 k - 1\leq 2 \log_2 d$. If
$$ 3\ 2^k -3 \leq d'\leq 6\ 2^k -3$$
then it is possible to construct a commensurate path overcoming the
distance $d'$ simply by doubling {\it some\/} of the
steps in the sequence above or possibly by tripling the middle step. We
need at most $2 k - 1 + k+2 \, \leq\, 3 \log_2 d' \, $
steps which completes the proof.
\enddemo
Now we come to the definition of a {\it cluster\/} :
\definition{Definition 3}
The notion of a {\it cluster\/} of sets (only {\it some
\/} sets will be employed in the construction of
clusters, see below) is defined recursively, retaining the
letter T for the notation of clusters, as follows:
\roster
\item "{i)}" Any set
$$ T = \supp \gb$$
where $\gb$ is a contour or an admissible system (to be specified
below; we will consider below only some special, ``recolorable'' systems $\gb$
which will be defined later) is a cluster.
\item "ii)" If $T_i$ are some clusters and $T_0$ is from i) such that
$$ \dist (\supp T_0, \supp T_i) \leq \text{min}\{ \diam\supp T_0, 2\ \diam
(\supp T_i)\}
\tag 3.5 $$
holds for each $i\geq 1$ then the collection
$$ T= (T _0, \{T_i\})$$
is again a cluster. We denote
$$ \supp T = \supp T_0 \cup \cup _i \supp T_i\,\,. $$
The set $T_0$ will be called the {\it core\/} of the cluster $T$.
\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 appearance of the additional ``logdiam'' term
(compared to $\Conn T$) in the definition
of $\conn T$
will be seen to be related to our formulation of the condition
(3.5). See Proposition below.
\definition{Definition 4}
We assign, to any cluster $T$, a commensurate tree $\Cal T$ as follows:\ If
the trees \ $\Cal T_0$ and \ $\Cal T_i$ \ are already constructed -- by
Definition 2
and 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)
all the trees \
$\Cal T_0$ and \ $\Cal T_i$
and such that all branches of \ $\Cal T \setminus \Cal T_0$ start with
some loop of the type $\{t',t''\}$.
%(This is a suitable technical
%requirement enabling to reconstruct back
%the ``core '' $\Cal T_0$ as well
%as the subtrees $\Cal T_i$ uniquely from $ \Cal T$.)
% The root of $\Cal T_0$ is proclaimed
%to be also the root of $\Cal T$.
%The supports of \ $\Cal T_i$ and of \ $\Cal T_0$ are all considered to be
%mutually disjoint (by taking additional ``copies'' of the same $t\in \zv$
%if necessary).
To have an idea about the length of $\Cal T$ consider a tree
$$ \Cal T' =\Cal T_0\, \cup \, \cup_i (\Cal T_i \cup P_i)$$
where $P_i$ are some shortest possible commensurate paths, each of them
starting
in some loop $\{s_i, s'_i\}$ of $\Cal T_0$ and ending in some loop
$\{t_i, t'_i\}$ of $\Cal T_i$.
%The supports of $\Cal T_i$ and of \ $\Cal T_0$ are again all considered to be
%mutually disjoint and the same is assumed for the connecting paths $P_i$
% -- except of their ``starting points'' $s_i$, of course.
\newline
In analogy to Definition 2, the quantity $\Conn T$
is now defined as the length of the tree $\Cal T $ and we put
$$ \conn T = \Conn T + [ 3 \nu \log_2 \diam \supp T ] . \tag 3.6 $$
\enddefinition
In order to reconstruct back the original cluster $T$ from a given tree
$\Cal T$
the following notion will be useful:
\definition{Definition 5}
Assume that some total ordering $\prec$ on the collection of all
subsets of $\zv$ is defined, extending both the lexicographic order
between the shifts of a given set as well as the relation
$A \prec B$ if $A\subset B$.
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 in the recurrent
definition of the clusters $T_i$.
\enddefinition
\remark{Notes}
1) Notice that $\Conn T $ is not greater than $|\Cal T'|$ \ i.e.
$$ \conn T \leq |\Conn T_0| + \sum_i |\Conn T_i| + \sum_i l_i +
[ 3 \nu\log_2 \diam \supp T ]. \tag 3.6'$$
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 rewrite the present section for
this (narrower) setting in the spirit analogous to that of the
later Theorem 6,
without employing the bothering (but small!)
logdiam terms. We prefer the more general exposition here in view
of wider applicability of the estimates obtained here also to other situations.
\newline
2) The mapping from clusters to trees constructed above
is not one to one. However, if we assign to any bond of $\Cal T$
a ``flag'' i.e. mark it by a value $0$ or $1$
and interpret the components of $\Cal T$ marked by $0$ resp.
$1$ as the sets used in the definition of $T$ resp. as the connecting
paths (connecting $T_0$ with $T_i$ etc.) then the core of the cluster $T$
(and the cores of $T_i$ etc.)
can be recognized just by looking at the biggest (at $\prec$)
$0$ -- component of $\Cal T$.
Thus, one has a crude bound $2^n$ for the number of standard
clusters $T$
with the same tree $\Cal T$ of the cardinality $n$.
In the forthcoming applications, all clusters constructed by us will be
standard ones and so we will not discuss the possible modifications of the
estimates discussed below which would be
needed if also nonstandard clusters would appear.
\endremark
The following estimate will be used later in Theorems 5 and 5' (though in
a slightly changed form). It says that having established a
slightly stronger version of the estimate (3.1) for the {\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 here.
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 --
which will be possibly interesting
also in other situations where
our method can be applied. This latter result
resembles the classical Meyer method
(see Ruelle's book \cite{R1}).
\proclaim{Proposition 1}
Assume that the quantities $k_T$ are defined recursively by formulas
$$ k_T = k_{T_0} \prod_i k_{T_i}. \tag 3.7 $$
Assume that for the {\it sets\/} $T_0$ the following stronger variant of (3.1)
is valid:
$$ | k_{T_0} | \leq \varepsilon ^{\conn T_0 +6 \nu\log_2 (\diam T_0 + 6)}
. \tag 3.8 $$
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) from which the
clusters $T$ were formed.
\endproclaim
\demo{Proof}
It suffices to prove that
$$ \Conn T_0 + 6 \nu\log_2 (\diam T_0 +6)+ \sum_i \conn T_i \geq \conn T
. \tag 3.10 $$
Notice first the following simple estimate. Define the support $\supp T$
and
the diameter
$\diam T$ of a cluster $T=\{T_i\}$ recursively by
putting $\supp T = \supp T_0 \cap \supp T_i$
and $\diam T = \diam \supp T$.
\proclaim{Lemma 2}
Let $T_j$ be the longest of all clusters $T_i$
(maximizing its diameter). Then
$$ \log_2 \diam T \leq \log_2 (\diam T_0 +6) + \log_2\diam T_j$$
assuming that the right hand side is greater or equal to 6.
\endproclaim
\demo{Proof of Lemma 2}
It is straightforward: notice that the condition
(3.5) implies the bound
$$ \diam T \leq \diam T_0 + (1+2+2+1) \diam T_j . $$
Then we use the inequality $\log_2 (x + 6y) \leq \log_2 (x+6) + \log_2 y$
which is surely valid if $ x \geq 1 \text{ and }
y \geq 1$.
\enddemo
Notice that it suffices now to establish
the following bound (from which the required bound (3.9) is obtained
by summing with the bound of Lemma 2):
$$ \Conn T_0 + 3 \nu\log_2 (\diam T_0 +6) + \sum_{i \neq j} \conn T_i
+ \Conn T_j \geq \Conn T $$
which is surely valid because
then we can rewrite it (notice that $l_j \leq 3 \nu\log_2 (\diam T_0 +6)$
by Lemma 1 and omit the number 6! ) in a stronger form
$$ \Conn T_0 + \sum_i (\Conn T_i + l_i) \geq \Conn T
$$
where $l_i$ 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 $ \ follows from the condition (3.5).
The validity of the last inequality follows from the very definition
of $\Conn T$ (see (3.6')) and this completes the proof of Proposition.
\enddemo
\remark{Notations}
In the following we will usually write, for clusters $T$,
$$ t\in T, T \subset \Lambda, \dist(T, \Lambda), \dots$$
instead of the more precise notations
$$t\in \supp T, \supp T \subset \Lambda, \dist( \supp T,\Lambda) \dots. $$
Writing $G \in T$ we will mark
the situation when the set $G$ was used in the recursive definition of $T$
(as the ``core'' of some intermediate cluster used in the construction)
of the cluster $T$.
\endremark
Finally we formulate one 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 $T$,
$$ |k_T| \leq \varepsilon^{\conn T} \tag 3.11 $$
then the cluster
series with the terms $k_T$ quickly converge in the following sense:
for any
$t \in \zet$ and for any $d \in \en$ we have
$$ \sum_{T; t \in T, \conn T
\geq d} |k_T| \leq (C\varepsilon)^d \tag 3.12 $$
and analogously for $\conn T \geq d$ replaced by $|T| \geq d$.
\endproclaim
First notice that instead of $\conn T$ it suffices to prove an analogous
result for the smaller and
``more natural''
quantity $\Conn T$. We 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
(3.9) which should be assumed to be valid for all {\it sets} $T$. Once
we have (3.12) 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 the following broader setting (which
is closely related to usual estimates in the theory of the
Mayer expansions -- see the book [R1]).
It is easy to understand that Proposition 3 below
actually {\it generalizes\/} the statement of Proposition 2:
Recall that we identified any cluster $T$ with some
commensurate tree
$\Cal T$ on $\zv$ and the number of standard clusters corresponding
to $\Cal T$ is at most $2^{|\Cal T|}$. Having this in mind the
forthcoming result can
be formulated for quantities $ k_{\Cal T}$ indexed by
commensurate {\it trees\/} $\Cal T$ on $\zv$.
% The mapping \{ cluster $\to $ tree \} is injective i.e.
% two different clusters are always mapped to different
% trees. Recall that the core $T_0$ of the cluster $T$
% is always identifiable from the corresponding tree(having a marked root):
% it is the largest subtree
% containing any site of the lattice $\zv$ at most once!
\proclaim{Proposition 3}
Let the quantities $k_{\Cal T}$ be given as products of some quantities
denoted by $k_{\{t,s\}}$ or $k_b$ (see the commentary below)
$$ k_{\Cal T} = \prod_b k_{\,b} $$
where the product is
over all the ``bonds'' $b = \{\{A,B\},\phi\}$
of the commensurate tree $\Cal T =\{G,\phi\}$.
Recall that the ``bonds''
$\{A,B\}$ of an abstract tree $G$ are mapped by
$\phi$ to unoriented pairs $\{t =\phi(A),s =\phi(B)\}$ of points of $\zv$.
The notation $k_b$ is used instead of a more explicit
notation $k_{\{t,s\}}$ for $b =\{\{A,B\},\phi\}$ such
that $\{t =\phi(A),s =\phi(B)\}$.
Assume that these quantities $k_b$ are nonnegative and $k_{\{t,t\}} = 1 $
for each $t$.
Let for any unordered pair $ b = (t,s) $ (a slight abuse of notations)
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 also the bound
$$ \sum k_{\Cal T} \leq k_b \cdot q' \tag 3.14$$
where the
summation in (3.14) is over all commensurate trees $\Cal T$
containing the ``bond'' $b$ as its ``extremal bond'' and having a length
at least $2$. By the extremality
of \ $b = \{t =\phi(A),s =\phi(B)\}$
we mean here that
one vertice of the pair $ \{A,B\}$
remains ``free'',``endvertice'' in the original abstract
tree; this must hold at least for one of the bonds $\{A,B\}$
which are mapped to the
given pair $ \{t,s\}$.
% commensurate with respect to the given bond $b$ (i.e. over trees $\Cal T$
% such that
% $ the tree $\Cal T \cup b$ is also commensurate).
The quantity $q' $ can be chosen like $3q$. \endproclaim
\remark{Note} More precisely, the optimal value of $q'$ can be found
from the equation (see the end of the proof below)
$$ \exp( q'') = 1 +q' $$
where $ q''$ denotes the supremum, over all bonds $b$,
of the sums analogous to that in the left
hand side of (3.13)
but with modified terms
$$ k_{b'}''= { k_{b'}(1+q')\over 1- k_{b'}(1+q')} $$
where $q'$ has its previously established value.
Namely if all $k_b$ are small and we already have established
the smallness of $q'$ then we have the crude
bound $k_{b'}'' < 2 k_b (1+q')$; hence we have also the
inequality $q'' < 2q(1+q')$. See the end of the proof below.
\endremark
\demo{Proof} Apply the method of induction. Denote by $\sum^{ 0\,$
such that for any $T$,
$$
|k_T|\leq \varepsilon ^{\,\conn T} .
\tag 3.17
$$
\endroster
\remark{Notes} \newline
{\bf 1. }
%We will really need to consider also the boundary
%conditions $y$ which are only {\it locally\/} but not
%{\it globally\/} from $\es$.
%Imagine a set $ \Lambda $ which is `` not horizontally
%connected'' -- like some ``bumerang'' whose corners are in
%the same horizontal level.
%(Such a bumerang can appear as the ``interior'' of some
%contour.)
%Then the configuration $y$ which is locally from $\es$
%can have quite different values on different corners!
%\newline
%{\bf 2. }
We will later glue together -- in our ``recoloring
procedures'' -- contours $\gb$ and clusters $T$ such that
$\dist(T, \supp\gb)\leq 2\diam T$ -- 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'' $3 \ \log_2 \diam T$
to the quantity $\Conn T$ in the definition of $\conn T$
to keep the control over the connectivity properties
of the new clusters formed by such (recursive) procedures.
\newline
{\bf 2. }
The collection of allowed contours (and of allowed
configurations, see below) will vary from
one mixed model to another. Typically the allowed set of contours
will be some
subset of the original collection
of contours (of some given abstract Pirogov -- Sinai model)
-- and this subset will become even {\it smaller\/}
after 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 3. }
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 with
(``slightly'') translation noninvariant models; the new cluster
quantities $k_T$ will be constructed successively in the lexicographic order,
through an infinite sequence of
intermediate (noninvariant) mixed models.
However, 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 admissible systems are allowed in $\zv$) we will speak about
the {\it translation invariant\/} mixed model.
\newline
{\bf 4. }
Having specified the collection of allowed {\it contours\/} of the mixed model
we do not even require that all admissible collections of allowed
contours are allowed configurations of the mixed model. At this moment
we impose no special requirements on what collections of contours
are really allowed in our model; see the forthcoming sections 3,4,7
for a more concrete information about the actual choice
of the configuration space.
\endremark
\vskip 5mm
The {\it partition functions\/} of the mixed model in a given volume $\Lambda$
({\it strictly\/} diluted ones; we can forget the notion of a
diluted partition function for most of part III; however our volumes
$\Lambda$ will be often the standard and moreover the conoidal ones)
will be given as
$$
Z^{\alpha}_{\updownarrow}(\Lambda)=
\sum_{\Cal D \subset \subset \Lambda} Z^{\alpha }_{\Cal D }
(\Lambda )
$$
where $\alpha $ is a boundary condition on $\partial \Lambda^c$
(which is locally from $\es$) and $\Cal D $ is an admissible
family of contours; the notation $\Cal D \subset \subset
\Lambda $ will mean, everywhere in the following,
that $\dist(\vv), \Lambda ^c)\geq 2$
for any contour $\gb$ on $\Cal D $. By (3.15) we define
$$
Z^{\alpha }_{\Cal D }(\Lambda )=
\exp(-\sum_i E(\gb_i))
\exp(-\sum_{t\in\Lambda }e_t(\delta ))
\exp(\sum_{T\subset \Lambda \setminus\supp\Cal D
}k_T(\delta))
\tag 3.18
$$
where $\delta $ denotes the configuration$(\alpha
\cup\partial \Cal D )^{\text{best}}$.
\remark{Note}
It seems unnatural to use the symbol $Z$ for the ``mere Gibbs
factor'' (3.18). However, the case $k_T = 0$ is not a typical
example here. In a more general case, the considered mixed model
corresponds actually to some partial expansion of the model (2.18).
Then (3.18) 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 a single internal admissible subsystem $\gb$
\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)
modification 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.
Recoloring will be just one step towards the desired
``total expansion''
of the model, and this step is described in detail in Lemma
and Theorem 4 below.
Roughly speaking, recoloring of $\gb$ 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 and where some {\it new\/}
quantities $k_T$ (for some {\it new} clusters $T$ containing
$G$ 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 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 to (2.21);
see also the remark below Definition 2.
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.
Let us introduce again (it will be quite indispensable in what
follows) the definition of a suitable ``protecting zone''
of the volume $V(\gb)$:
\definition{Definition 1}
Define first the ``conoidal'' volume
$$
\vv=V(\gb)\cup V'(\gb)
\tag 3.19 $$
where $V'(\gb)$ is the collection of all points $t\in(V(\gb))^c$
satisfying the bound
$$
\dist(t, V(\gb)) \leq \frac{1}{2}\dist(t, \supp \gb)\, .
\tag 3.20
$$
Denote by $A(\gb)$ , more precisely by $ A(\gb,x_{\vv})$ the quantity
$$
A(\gb,x_{\vv}) =
\sum_{t\in V(\gb)}(e_t(y)-e_t(x)) +
\sum\Sb T\subset \vv\\T\cap\supp\gb=\emptyset\endSb
k_T(x_{\vv})-
\sum_{T\subset \vv}k_T(y)
\tag 3.21
$$
where $x=(\partial \gb)^{\text{best}}$ (the configuration
minimizing the hamiltonian under the condition $\partial \gb$)
and $y=x_{\vv}^{\ext}$.
\enddefinition
\remark{Note}
Thus, to be able to define $A(\gb)$ 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 $T$ and we do not assume $T\subset V(\gb)$).
However, the value of $x$ on $\vv$ will be normally
determined by the context in which the contour (or
admissible system) $\gb$ will appear and so we will 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
$$
F(\gb)=E(\gb)-A(\gb)
\tag 3.22
$$
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''(which would not be the case if we would take
mere $V(\gb)$ here).
However, if $\gb$ is an interior subsystem of some bigger
admissible collection $\gb\&\Cal D $ of all contours of
some configuration $(x_{\Lambda }, \gb\&\Cal D )$ in a finite
volume $\Lambda$, the following modifications of the
quantity $A(\gb)$ will be sometimes considered later, too:
(We will be interested below only in the cases when moreover
$\vv \cap \Cal D = \emptyset $ i.e when $\gb$
is not ``too tightly attached'' to $\Cal D$.)
\definition{Definition 2}
In analogy to (3.21) define also the modified quantities
%$A_{\Cal D ,\ \Lambda }(\gb)$,\
$A_{\loc}(\gb)$,\
$A_{\full}(\gb)$,
$A_{\full,\Cal D, \Lambda }(\gb)$\
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
%$(\vv\cap\Lambda )\setminus \supp\Cal D $,\
$V(\gb)$,\ $\zv$,\ % $\Lambda \cap \vv$,
$\Lambda \setminus \supp\Cal D $. \
Corespondingly define, by (3.22), the quantities
%$F_{\Cal D , \Lambda }(\gb)$ and
$F_{\loc}(\gb)$,\
$F_{\full }(\gb)$,\
%$F_{\Lambda }(\gb)$,\
$F_{\full,\Cal D , \Lambda }(\gb)$
and also $F_0(\gb)$) (taking $A_0(\gb)$ from (2.20)).
\enddefinition
\remark{Note} Assuming the existence of 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 be inpractical in the expansions
constructed below. However, we 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 a ``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
\footnote{ A general ``philosophical'' remark: sometimes, one is fighting severe
technical problems in the \ps \
theory which however
start to be relevant only in volumes
which are really {\it astronomically large\/}; for example the problem
mentioned above is hardly of much relevance in volumes
of a size, say $10^{27}$!}
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 O.K.
because contours are ``crusted'' in that case.
The quantity $F(\gb)$ %(resp. $F_{\Lambda}(\gb)$ in the vicinity
%of $\Lambda^c$)
is a reasonable compromise
because it approximates \ $F_{\full }(\gb)$ \ with a great
accuracy $(\sim \varepsilon |\supp \gb|)$ and at the same
time it is ``local'' in a reasonable sense. One could take
even smaller sets $V'(\gb)$ to retain this accuracy but our
choice will have additional advantages below in Theorem 5.
The quantity $F_{\full,\Cal D , \Lambda }(\gb)$ is just
a temporary notation used in the
proof of Lemma below.
\endremark
The forthcoming lemma is an {\it essential step\/} in the procedure
called ``recoloring of an internal contour''. This is further
developed by Theorem 4, concluding the effort of Section 3.
\proclaim {Lemma}
Assume that we have a mixed model satisfying (3.1).
Let $\Lambda \subset \zv$. Let $\Cal D\ \&\ \gb\subset \subset
\Lambda $ be an admissible system such that $\gb$ is its
removable subsystem, satisfying moreover the condition
$ \vv \cap \Cal D \cap \Lambda^c =\emptyset$.
Let $\alpha $ be a boundary condition on
$\partial \Lambda ^c$ which is in conformity with $\Cal D\ \&\ \gb$
(such that there exist a configuration
$x_{\Cal D\&\gb}=(\alpha \cup \partial \Cal D \cup \partial
\gb)^{\text{best}}$ for which all points of $(\supp \Cal D
\cup \supp\gb)^c\cap\Lambda $ are stratified).
Then
$$
Z_{\Cal D\&\gb}^{\alpha }(\Lambda )=
Z_{\Cal D}^{\alpha }(\Lambda )
\exp(-F(\gb))
\exp(\sum k_T^{\text{new}})
\tag 3.23
$$
where $k_T^{\text{new}}$ depends on both
$x_{\Cal D\&\gb}/T$ and $x_{\Cal D}/T$ and the summation
is over all
$T\subset \Lambda $ such that $T\not\subset \vv, T\cap V(\Gamma)\neq\emptyset
, T\cap \supp \Cal D = \emptyset$
and
$\dist(T, \supp\gb)\leq 2\diam T$.
The quantities
$k_T^{\text{new}} $ are given by formulas (3.26), (3.27) below and they satisfy
the estimate
$$ |k_T^{\text{new}}| \leq 2\,\varepsilon ^{\conn T} \tag 3.24$$
for each $T$.% One can write $F(\gb)$ in (3.23) if
% moreover the condition
%$ \vv \cap \cap \Lambda^c =\emptyset$ is fulfilled.
\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
$$
\aligned
&\exp(-E(\gb)-E(\Cal D ))
\exp(-\sum_{t\in\Lambda }e_t(\gamma \delta ))
\exp(\sum\Sb T\subset\Lambda \\
T\cap (\supp \Cal D \cup \supp\gb)=\emptyset
\endSb
k_T(\gamma \delta )
)=\\
&=\exp(-E(\Cal D ))
\exp(-\sum_{t\in\Lambda }e_t(\delta ))
\exp(-F_{\full,\Cal D , \Lambda }(\gb))
\exp(\sum\Sb T\subset\Lambda \\
T\cap (\supp \Cal D)=\emptyset
\endSb
k_T(\delta )
)=\\
&=Z_{\Cal D}^{\alpha }(\Lambda )
\exp(-F(\gb))
\exp(
\sum
\Sb T\subset \Lambda \\
T\not\subset \vv \\
T\cap(\supp\Cal D \cup \supp \gb)=\emptyset
\endSb
k_T(\gamma \delta )-
\sum
\Sb T\subset \Lambda \\
T\not\subset \vv \\
T\cap\supp\Cal D =\emptyset
\endSb
k_T(\delta )
)
\endaligned
\tag 3.25
$$
which proves (3.23) with the following choice of the
quantities $k_T^{\text{new}}$ (recall that the ``old''
quantities $k_T$ satisfy (3.1), hence we have (3.24) ):
$$
k_T^{\text{new}}=k_T(\gamma \delta ) - k_T(\delta )
\tag 3.26$$
\newline
$ \quad \text{if} \quad
T\subset \Lambda\, , \, \,
T\not\subset \vv\, , \, \,
T\cap(\supp\Cal D \cup \supp\gb)=\emptyset
,T\cap V(\Gammab) \neq \emptyset$, resp.
$$
k_T^{\text{new}}=-k_T( \delta )
\tag 3.27 $$
\newline
$ \quad \text{if} \quad
T\subset \Lambda\, , \, \,
T\not\subset \vv\, , \, T \cap \Cal D = \emptyset,\,
T\cap\supp\gb\ne\emptyset
$. The condition $ T\cap V(\Gammab) \neq \emptyset$
(in (3.26); contrast it to the condition $T\not\subset \vv$!)
follows from the observation that
$$
T\not\subset \vv\, \&\, \dist(T, \supp\gb) \geq 2\diam T
\Rightarrow
T\cap V(\gb)=\emptyset
$$
$$
\Rightarrow
(\gamma \delta )_T=\delta _T
\Rightarrow
k_T^{\text{new}}=0\, .
\tag 3.28 $$
(Compare the definition of $\vv$; the condition
$T\cap V(\gb) \ne \emptyset \ \& \
T\not\subset \vv$ would imply that there is some $t\in
T\setminus \vv$ with $\dist (t, V(\gb))\leq
\frac{1}{2}\dist(T, \supp \gb)$. )
\enddemo
\definition{Notation}
An interior subsystem $\gb$ of an admissible
system $\gb'$, satisfying
also the condition $\vv \cap (\gb' \setminus \gb) = \emptyset$
will be called a {\it strictly interior\/} subsystem of $\gb'$.
\enddefinition
\proclaim{Theorem 4}
Assume that we have a mixed model satisfying the condition (3.1).
Consider the partition functions ((3.18))
$$
Z_{\Cal D, G}^{\alpha}(\Lambda) =
\sum_{\gb:\supp\gb=G}Z_{\Cal D \&\gb}^{\alpha }(\Lambda)
$$
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 an admissible system such that $\gb$ is
its strictly interior subsystem, satisfying moreover the
condition $ \vv \cap \Lambda^c =\emptyset$.
(The partition functions above
correspond to the events ``\ $\Cal D \ \& \ \gb
$ appears'' resp. ``\ $\Cal D \ \& \text{ possibly }
\gb $ appears'' ; $\gb$ is a contour such that $\supp \gb = G$.)
These partition functions can be expressed as
$$
Z_{\Cal D, G }^{\alpha }(\Lambda )=
(\sum_{T:G\in T} k_T^+)Z_{\Cal D}^{\alpha}(\Lambda)
\tag 3.30
$$
resp.
$$ Z_{\Cal D, [G]}^{\alpha}(\Lambda)=\exp(\sum_{T:G\in T} k_T^*)
Z_{\Cal D}^{\alpha}(\Lambda)
\tag 3.31
$$
where $G\in T$ means that $G$ is the core of the cluster T
and $k_T^+$ resp. $k_T^*$ are some new cluster terms, described
in detail in the proof.
The leading new quantity $k_G^* = k_G^+$ is equal to
$$ k_G^+ = k_{G, \alpha}^+ = \sum_{\gb:\supp \gb = G} \exp
(-F(\gb)).
\tag 3.32 $$
The remaining quantities $k_T^+$ obey the bounds, for any cluster of the type
$T =(G, \{T_i\})$\hfil \break
(where $T_i \ ;i = 1, ..., m $ are clusters of the given mixed model)
$$ |k_T^+| \leq k_G^+ \ \varepsilon ^{\ \sum_{i=1}^m \conn T_i}\,
\tag 3.33 $$
and analogous bounds are valid for $k_T^* $.
See the proof ((3.41)) for the more precise form of the
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 the preceding Lemma; the
quantity $k_G^+$ does not already depend on the interior of $\gb$
in contrast to the quantity $\exp(-F(\gb))$.}
and are translation invariant in the sense of (3.15).
Moreover, if $\ k_G^+\ $ satisfies (3.7) then the validity of the bound
(3.1) in the given mixed model implies
its validity also for the new quantities
$k_T^+$ and $k_T^*\ $.
\footnote{ In the next section, we will formulate
a ``horizontally invariant''
modification of Theorem 4, namely Theorem 5.}
\endproclaim
The procedure called ``recoloring of a contour''
described by formula (3.31) will be used later repeatedly many times
for all the ``smallest possible'' strictly interior subsystems
$\gb$.
The new cluster quantities $k_T^*$ will play later the same role as the
``old'' quantities $k_T$); and therefore it will be crucial to ensure
the validity of the estimate (3.1) for them:
Comparing (3.33),(3.32) with (3.7) (Proposition of Section 1)
one sees that the sufficient smallness of
$k_G^*$ will be guaranteed by the following estimate :
\definition{Definition}
We will say that an admissible system $\gb$ is {\it recolorable\/}
%in $\Lambda$ (or, simply, recolorable if
% $\Lambda^c \cap \vv= \emptyset$)
if
$$ \sum_{\gb :\supp \gb = G}\exp (-F(\gb)) \leq \exp(-\tau'\conn G)
\tag 3.34 $$
holds with a sufficiently large $\tau'$ (to be specified in the
Corollary below; see (3.35)).
The constant $\tau'$ will be
chosen below roughly as $\frac {\tau}{ 12 \nu}$.
\enddefinition
\remark{Notes}
1. Notice that this is the requirement on the {\it set \/} $G$,
not on a particular contour $\gb$ -- though practically this is closely related
to
saying that for each $\gb$ with the same support $G$, one has a bound
$$ F(\gb) \geq \tau '' \conn \gb \tag 3.34''$$
with some other large $\tau''$.
\newline
2. One should always have in mind that the property ``to be
recolorable'' is rather sensitive (for contours of {\it very\/} large
size, of course) with respect to the boundary
conditions. The mere knowledge of the external
colour of $\gb$ (at the horizontal level of $V(\gb)$) may be insufficient to
decide the recolorability. In general, the external
colour of the whole $\vv$ must be known.
\endremark
\proclaim {Corollary}
Let $\gb$ be recolorable, with $\tau'$ satisfying the inequality
$$ \exp(-\tau'\conn G) \leq \varepsilon ^{\Conn T + 6\nu \log_2 (\diam T + 6)}.
\tag 3.35 $$
Then the validity of the bound (3.1) (with $\conn T$ defined by (3.6))
in the
given mixed model implies its validity also for the new quantities
$k_T^+$ and $k_T^*$.
\endproclaim
\remark{Notes}
This means that the new mixed model
constructed by the formula (3.31) (which has a new, richer family of
cluster fields $\{k_T\} \& \{k_T^*\}$ but which does not yet allow
$\gb$ as an interior subsystem of its configuration)
is of the {\it same type\/} as before, satisfying again the estimate
(3.1).
The bound (3.35) relates the appropriate choices of constants
$\varepsilon, \tau'$\ in (3.1) and (3.34). The latter condition
is a Peierls type condition for the quantity $F(\gb)$ , and one has
to choose a suitable $ \tau ' \leq \tau$ there, to obtain a useful
notion. ($\tau$ is from (2.17).)
This will be discussed later and we will see that when taking
$\tau' = c \tau $ the convenient choice of the constant will be $c=
\frac {1}{12\nu} $.
One should also notice that later, when performing 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}
By (3.23) we have
$$
Z_{\Cal D, G }^{\alpha }(\Lambda)=
\sum_{\gb:\supp\gb=G} Z_{\Cal D \&\gb}^{\alpha }(\Lambda)=
Z_{\Cal D }^{\alpha }(\Lambda) \sum_{\gb}
\exp(-F(\gb))
\exp(\sum_T k_T^{\text{new}})\, .$$
Writing $\exp(-F(\gb)) = \xi _{\gb} k_G^+$ and expanding the
exponential this can be written as
$$ Z_{\Cal D, G }^{\alpha }(\Lambda)
= Z_{\Cal D }^{\alpha }(\Lambda) \biggl(\sum_{\gb} \xi _{\gb}
\sum_{k=0}^{\infty} \sum\Sb (T_1, \dots, T_k)\\
(n_1, \dots, n_k) \endSb
\prod_{i=1}^{k}\frac{1}{n_i!}(k_{T_i}^{\text{new}})^{n_i})\biggr)
k_G^+ =
Z_{\Cal D }^{\alpha }(\Lambda)(\sum_T k_T^+)
\tag 3.36$$
where the new values of $k_T^{\text{new}}$
are denoted here as
\footnote{It is perhaps worth mentioning that whereas
$F(\gb)$ depend on the values $\gb$, in particular on the
``interior colour of $\gb$'', the quantity $k^+_T$ already
depends only on $G$
and on the ``external colour'' of $\gb$.}
$$
k_T^+ = \sum\Sb \gb:\supp \gb = G\endSb
\xi _{\gb} k^+_G \prod_{i=1}^{k}\frac{1}{n_i!}(k_{T_i}^{\text{new}})^{n_i}
. \tag 3.37$$
Here, the clusters $T$ are defined, for $n_i=1$, as
$$ T= (G, \{T_i\}) . \tag 3.38 $$
(including also the empty collection $\{T_i\}$); otherwise the
cluster $T$ contains the corresponding number $n_i$ of copies of $T_i$.
Notice that the expression of $k_T^+$ by values $k_T^{\text{new}}$
is not exactly as (3.7), Proposition of Section 1.
However, it is straightforward
to adapt the corresponding estimates noticing that $\sum_{\gb} \xi _{\gb}
= 1$. Thus, one obtains (3.33) and then, from (3.34) and (3.35),
also (3.1) for the
quantity $k_T^+$.
(If some $n_i > 1$ then the estimate is even a stronger one).
The expansion (3.31) is obtained by taking logarithms:
$$
\log Z_{\Cal D, [G]}^{\alpha}(\Lambda) = \log Z_{\Cal D}^{\alpha}(\Lambda)+
\log(1+ \sum_{T:G \in T} k_T^+) =
\log Z_{\Cal D }^{\alpha }(\Lambda)+ \sum_{T:G\in T} k_T^*
\tag 3.39 $$
where
$$
k_T^*= \frac{(-1)^{n-1}} {n}
\prod_{i=1}^{n}k_{T_i}^+
\tag 3.40$$
for any cluster $T=(T_1, T_2, \dots, T_n) $. (Again,one has to modify
correspondingly this formula if multiple copies of one cluster
$T_i$ appear in $T$).
%This
% concludes the proof of Theorem 4 and of its Corollary.
Strictly speaking, clusters of such a type were not defined yet.
However, if e. g. three clusters $T_1$ , $T_2$ and $ T_2$ have the same
core $G$ then
we identify the ordered triple $(T_1, T_2,T_3)$ for the
clusters $T_i$ given as
$T_i =\{G,T^*_i\} $ ( where $T^*_i$ denotes some collection of
clusters) with the cluster
$$ \{G, T^*_1,\{G,T^*_2,\{G,T^*_3\}\}\}. $$
Now, for any
such cluster $T= (T_1, T_2, \dots, T_n)$ (recall that $G\in T_i$ for
each $ i$!)
one obtains, after some inspection, the desired bound
$$ |k_T^*| \leq (k_G^+)^n\ \varepsilon ^{\sum_{i=1}^n \conn ^+ (T_i)}
\leq \varepsilon ^{\conn T}\tag 3.41$$
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 (3.37) above.
This
concludes the proof of Theorem 4 and of its Corollary.
\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 subsystems''.
Recall the ordering $\prec$ of systems of contours we introduced in section II
(see (2.11) and the text below it).
We will say that a recolorable system $\gb$ of contours is {\it smallest
recolorable} system of the given mixed model
if it is strictly interior (in the considered volume
$\Lambda$), recolorable and moreover there is no smaller
strictly interior
recolorable (smaller in $\prec$) $\gb'$ which would 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
contours $\gb$ 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 P. S.
abstract model; the configuration spaces of the mixed models thus
obtained will be defined in terms of
requirements on the size of the smallest interior recolorable subsystems
of the configuration. See the forthcoming sections for details.
\endremark
\definition{Equivalent mixed models}
Two mixed models will be said to be equivalent if
all their strictly diluted partition functions are the {\it same\/}.
(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.)
\enddefinition
\proclaim{Theorem 5}
Assume that we have a horizontally translation invariant
mixed model satisfying the condition (3.1).
Let $\gb$ be a recolorable \footnote{ smallest
possible (in a geometrical sense); this will be the case needed in the
applications of Theorem 5 below} subsystem
whose horizontal shift can appear as a strictly interior subsystem
of some configuration of the model.
Assume that $k_T = 0 $ holds for all clusters $T$ containing a shift of
$\supp \gb$
(``containing'' in the sense that
$\supp \gb$ was used in the recursive construction of the cluster $T$ ).
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 $\gb$
among its (smallest possible) strictly interior recolorable subsystems.
\item If $T$ is a cluster not containing $ \gb$
then the value of $k_T$ in the new mixed model is the same as before.
\item If $T$ contains $\gb$ then the new value of $k_T$
satisfies the condition (3.1), too, assuming that $\varepsilon$ and
$\tau'$ are such that (3.35) holds.
\endroster
\endproclaim
\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''
which has the configuration space defined by the requirement
that exactly those congigurations of the original mixed model are allowed
for which {\it no\/} interior subsystems $ \gb + t'$ such that $ t' \prec t $
exist.
If $s$ is the nearest greater point to $t$ ($ t \prec s $) in the
given volume $\Lambda$ (remember that we always work in finite volumes)
then we define the transition to the ``$s$ -- th model'' by the very
procedure
described in Theorem 4.
It is straightforward to check the translation
invariance (3.16) and unicity of the definition (not depending on
the actual volume $\Lambda$ if $T$ is distant from
its complement -- in the sense of (3.16)) of the new
quantities $k_T$ thus obtained.
\enddemo
\remark{Note} There are also other methods 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 Lemma and Theorem 4
above for a {\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 here, in this paper.
Such an approach is used also in the recent lectures \cite{ZRO}
(in a simplest possible form, we believe)
and in future, we plan to replace the arguments based on the successsive
use of the lexicographic order by this more standard approach.
\endremark
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\head 5. Small and extremally small systems of contours
\endhead
We are still working with a general mixed
model. Only later we will explain the relevant choice of a
mixed model in various concrete situations ; this choice
will always be given as some
partial expansion of the original abstract Pirogov -- Sinai model
which we are investigating.
The successive application of the recoloring procedures
constructed in the preceding chapter will finally lead
to a family (indexed, in any volume, by elements of $\es$)
of mixed models where {\it no\/} recolorable systems will
be left! The reader is advised to skip 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 subsystem -- and the related
notion of a small resp. extremally small subsystem introduced
in this section -- in more depth, to obtain a useful supplementary
``topological'' result (Theorems 6 and 7) needed in the proof
of the Main Theorem.
Our discussion of the forthcoming notions of a ``small'' and
an ``extremally small'' system, and the notion of
a ``skeleton'' introduced below (notice that we are introducing
there another testing quantity $A^*(\square)$ -- as a more
careful alternative to $A(\gb) $) is
perhaps slightly more detailed than absolutely necessary
(if a shortest possible proof of Main Theorem
is required).
However, we are keeping it here as we expect our more
detailed exposition to be useful not only here
(giving more information
and estimates with better constants) but also
in the investigation of the ``metastability'' problem and in the study of the
completeness of the phase picture constructed by
our Main Theorem.% (See the concluding notes at the end of this paper.)
The reader interested in acquiring the idea of the proof
of Main Theorem can skip now directly to section 8 -- omitting even the
very notion of a small contour (given below) but finally realizing that
some variant of such a notion
(and the topological Theorem 7) is needed there!
Recall the definition of $A(\gb)$, $A_{\text{full}}(\gb)$, $A_{\text{loc}}
(\gb)$ and of
the corresponding quantities $F(\gb)$
from section 3, definition 2.
When checking the condition (3.34) (through (3.34''))
one needs
inequalities of the type
$$ A(\gb) \leq \tilde \tau |\supp \gb | $$
where $\tilde \tau $ is a suitable, ``not too large'' constant
(like $\tau \over 2$).
In fact, there is quite a freedom in the choice of the
constants $\tilde \tau,\ \tau', \ \tau''$ (from (3.34)
resp. from the relation (3.49) below) and
the difference between the various variants
of $A(\gb)$ resp. $F(\gb)$ (which was so
important in Section 3) will be quite irrelevant here
\footnote{except of the choice of
$A_{\text {loc}}(\gb)$ -- which would be too rough in some situations
where some ``reallly big'' ceilings appear.}.
Namely, we obviously have the
following bound.
\proclaim{Proposition}
$$ | A(\gb) - A_{\text{full}}(\gb)| \leq \varepsilon' |\supp \gb| \tag 3.42
$$
where the constant
$\varepsilon'$ is of the order
$\varepsilon^n$, $n$ being the cardinality of a smallest possible cluster
appearing in (3.21) and $\varepsilon = C \exp(-\tau)$ for a suitable
constant $C =C(\nu)$.
\endproclaim
The quantities $ A_{\text {full}}(\gb) $ (and, even more
importantly, the quantities $\Af(\square)$ introduced below) will
be more
convenient in these final sections than $A(\gb)$ and the
bounds of the type $ A(\gb) \leq \tilde \tau |\supp \gb | $
will be studied for them instead of $A(\gb)$.
Then we supplement these bounds by (3.42).
\footnote{The quantities $A(\gb)$ remain, of course,
in (3.22) but their test ``whether
they are dangerously big'' will now 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 found. One could ask this question
in a more precise way: whether the regime which resides inside $\Gammab$
is the ``best'' possible %one -- under given boudary conditions outside
and also what is the
``energetically optimal realization''
of such a contour.
Fortunately, one does not need to investigate
these questions in more detail, in particular the question
``what is the optimal shape of a contour'' is quite irrelevant here.
On the other hand, the question ``what is
the best 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{Definition}
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
configuration
$x$, define $f_t(x) =f_t(x_t^{\text{hor}})$ where $x_t^{\text{hor}}$ 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
$$\hat f_t(y) = e_t(y) - \sum_{t\in T} k_T(y) \tag 3.44 $$
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{ Namely,
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, or in
the interior of some bigger contour)
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 prescribed (fixed for once) way
to a configuration on the whole $\zv$.
The details of the
extension will be irrelevant.
\endremark
%
%
%\proclaim{Proposition}
%For any any finite volume $\Lambda$ we have the estimate:
%$$| \sum_{T\subset \Lambda} k_T(x) - \sum_{t\in \Lambda} (f_t^{\Lambda}(x)
%-e_t(x))|
%\leq \varepsilon' \ | \partial \Lambda| . \tag 3.44 $$
% \endproclaim
%For contours, or general admissible systems $\gb$ we
%(rephrasing once again (4.42)):
\proclaim{Proposition}
The quantity $A_{\text{full}}(\gb)$ can be expressed also by the formula
$$ A_{\text{full}}(\gb) = \sum_{t \in \zv} (f_t(x_{\gb}^{\text{ext}}) -
f_t(x_{\gb})) +\Delta(\gb) \
\ \ \text{where} \ \ \
|\Delta (\gb)| \leq \varepsilon' |\supp \gb| . \tag 3.45 $$
\endproclaim
\demo{Proof}
This follows
from
the observation that any $T$
``not touching $\supp \gb$'' which is counted in the definition of
$A_{\text{full}}(\gb)$ is counted also (exactly once!) in the above
sum $\sum_{t \in \zv} (f_t(x_{\gb}^{\text{ext}})-f_t(x_{\gb}))$.
The corrections to this observation (they are needed
only for clusters touching $\supp \gb$)
form, when summed together, the very term $\Delta(\gb)$
which therefore
satisfies (being the sum of small and quickly decaying
quantities $k_T |T|^{-1}$) the bound above.
\enddemo
The forthcoming notion will be useful for understanding
what would be the ``best possible gain in free energy''
inside a given contour $\gb$ :
\definition{Definition}
Given a configuration $x$ which is stratified outside of some
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 $V$.
We will usually consider such a configuration
$x$ in a volume $V =V(\gb)$, where $\gb$ is a contour or an admissible system.
\enddefinition
Notice that here, in comparison to the formulation of the
Peierls condition in Part II, we
use
the quantity $f_t(x')$ instead of $e_t(x')$. 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| $ which is a
quantity quite negligible when checking the validity
of the Peierls condition.
Now we are able to define an alternative
(with the same intuitive meaning)
to $A(\gb)$ -- which will
be more flexible in the fortcoming estimates.
\definition{Definition}
Given any finite volume \footnote{We will use later this quantity not only
for volumes $V= V(\gb)$ but also for
{\it cubes\/} $V$.} $V$ and any $y \in \es$
introduce the quantity
$A^*(V)$, more precisely $A^*_{\text{full}}(V,y)$ : %given by
$$ A^*_{\text{full}}(V,y)
= \max\{ \sum_{t \in \zv} (f_t(y) -f_t(z))\}
=\sum_{t \in \zv} (f_t(y)-f_t(y_{V}^{\text{best}}))
\tag 3.47$$
where the maximum is taken over all $z$
which are equal to $y$ outside of $V $ and where
$y_{V}^{\text{best}}$ is the configuration realizing this maximum.
%Analogously, using the quantities $f_t^{\Lambda}$ instead of $f_t$
%introduce the quantity $A^*_{\Lambda}(V,y) $ for any $y$ which
%is locally stratified on $\Lambda \setminus V^c$ and can be extended
%to some locally stratified configuration inside $V$.
\enddefinition
\remark{Notes}
0. Compare (3.47) with (3.45) (for $V= V(\gb)$).
We have, of course, the relation
$$ | A^*_{\text{full}}(V,y) - A(\gb)| \leq
\varepsilon' |\supp \gb| \tag 3.48$$
whenever $V= V(\gb)$
and $y$ is the external colour of $\gb$.
\newline
1. Our preference of this notion (to an alternative
$A_{\text{full}}(V)$
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 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 (typically, this energy will be of the order $\diam \square$ only)
just to compensate the ``volume gain inside $\square$\,'' (steming from
the fact that the configuration inside of such an artificial contour will
be assumed to ``jump freely to some better regime inside $\square$\,'').
The interplay between these formal notions and between 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 still some 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 $\Af(V(\gb))$.
In fact, it is advisable
\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
discussion of the ``dangerously big value of $A(\gb)$''
to the superordinated {\it cubes\/} only:
\endremark
\definition {Definition}
A cube $\square$ will be called {\it small\/} with respect to
a configuration $y \in \es$ if the following inequality
\footnote{Do not care about the particular choice of the
constant $\approx \tau$ 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 anywhere whenever the
energy of a contour
is considered)
would cause difficulties later.}
holds :
$$ A^*(\square,y) \leq \tau'\diam \square \tag 3.49 $$
where $\tau'$ is something like $ \tau' =\tau -\varepsilon'$
and $\varepsilon'$ is from (3.45),(3.48).
If $y$ is given only partially (on some neighborhood of $\Lambda$;
$y$ is, in fact, {\it always\/} given in some finite volume only)
then $\square$ will be called small with respect to $y$ if it is small for
{\it some\/} stratified extension of $y$.
Say that a cube $\square$ in $\zv$ is the {\it covering cube\/} of a volume
$S
\subset \zv $ resp. of a contour $\gb$
if $ \square$ is the smallest possible cube (smallest in the
usual ordering on the collection of cubes which is defined as an
extension of {\it both\/}
the lexicographic order of all the shifts of a single cube as well as
the inclusion relation between cubes)
which is a superset of $S$ resp.
of $\vv$. (Notice that we take $\vv$
instead of mere $V(\gb)$ here, the latter being
equivalent to $\supp \gb$ ).
We will denote the covering cube $\square$ by a symbol
$ \square(S)\ \text{resp.} \ \square(\gb)$.
If $\square$ is the covering cube of $\gb$ and
$y$ is the external colour of $\gb$ %(extended by some
%prescribed way to the whole $\zv$)
then we will say that
$\gb$ is {\it small\/} if $\square$ is small with respect to
$y$. We will say that a strictly interior subsystem $\gb$
of some admissible system $\Cal D$ in $\Lambda$, under some
boundary condition $y$ given of $\partial \Lambda^c$, is small in $\Lambda$
if
$\gb$ is small for the
exterior colour of $\gb$ induced by $\Cal D$ and $y$.
%More generally, a
%removable system $\gb $ of a configuration $(x, \Cal D )$ in $\Lambda$
%will be called small if
%$$ A^*_{\Lambda'}(\square \setminus \Lambda',x)
%\leq \tau \diam \square \ \ \ \
%\text{where} \ \Lambda' = \Lambda^c \cup \supp \Cal D . \tag 3.49
%$$
\enddefinition
\remark{Notes}
1. Recall that we take, here and everywhere in part III, the norm
$|t| = \max\{|t_i|\}$.\newline
2. The property ``to be small''
is formulated with the help of an
(arbitrarily chosen, but fixed)
extension of the external colour $y$ of $\square$ (extension to
the whole $\zv$). It will {\it not\/} be formulated for subsystems which
are not strictly interior in $\Lambda$.
%Noticing that the quantities $f_t$ in (3.47) are sums of
%exponentially decaying terms we see that the
% notion of a small subsystem $\gb$ of $x, \Cal D$ in $\Lambda$
% ``converges exponentially fast (with the distance
% $\dist(\supp \gb , \Lambda')$ )'' to the notion of a small $\gb$
% if the external colour of $\gb$ is extendable to some
% element of $\es$. \endremark
It is easy to see that for any small $\gb$
we have the inequality, with $y$ denoting the external colour
of $\gb$ (extended to the whole $\zv$, as mentioned above)
$$ A^*_{\text{full}}(V(\gb),y) \leq A^*_{\text
{full}}(\vv,y)
\leq A^*_{\text{full}}(\square(\gb))
\leq \tau' \diam \gb.
\tag 3.50 $$
Complement this with (3.42) and (3.45)!
The idea now is, roughly speaking, that all
small contours resp. admissible systems
should be
{\it recolorable\/}. This is obviously true
for {\it contours} because
we have from (3.50) and (3.42)
the following inequalities (see (2.17*), (2.8) and the Proposition
in part II, section 6)
$$ F(\gb) = E(\gb) - A(\gb) \geq (\tau^* -\varepsilon)|\supp \gb| -
\tau' \diam \gb
\geq {\tau^* \over 2} |\supp \gb| \tag 3.51 $$
and the last term is greater than, say $ \tau^* /4 \ \con \gb$
because contours are (as we know from Theorem 2, part II)
halfconnected.
However, we will often need to ``recolor'' also some {\it more
complicated\/} interior admissible systems $\gb$ with unclear apriori
relation between
$|\supp \gb|$ and $\con \gb$ (resp. $\conn \gb$). In such a case,
the corresponding more general argument (valid for any
admissible system $\gb$) will be developed in (3.54) below. However,
the notion of ``smallness'' has to be modified here
and the arguments (3.51) should be replaced by the
more detailed bounds
given below (in (3.60)).
\definition{Definition}
We will say that a small, strictly interior
subsystem $\gb$ of a configuration
$(x, \Cal D)$ is {\it extremally small\/}
in $\Cal D$ (more precisely in $(x,\Cal D)$ )
if it is small and moreover if the following recursive requirement for
$\gb$ is satisfied: for {\it no\/} strictly
interior subsystem $\gb' \subsetneqq \gb$,
$\gb'$ is already extremally small. %(with respect to $\Cal D \setminus \gb$;
%we do not mention explicitly the volume $\Lambda$ yet).
The recursion starts for those $\gb$ for which there are no strictly interior
$\gb' \subsetneqq \gb$ at all.
\enddefinition
\remark{Notes}
{\bf 1.}
This is the point where our interpretation of contours as supercontours
(recall that we replaced the original notion of a
contour by a more elaborate
notion of a supercontour already
in part II, section 7) finally becomes useful. We can claim now
that any admissible system $\Cal D$ has the following property:
after the removal of a %connected (in the relation $\to$)
removable subsystem $\gb$ from $\Cal D$, no
removable subsystem $\Cal D'$ of $\Cal D$
remains such that $\Cal D' \prec \gb$ but $ \gb \to \Cal D' $.
This property will be highly desirable when
recoloring the extremally small subsystems (which is something
which we will do later, when proving that extremal smallness
implies recolorability). Otherwise we could not use Theorem 5!
\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
even more 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.} %Given an admissible system $\Cal D$ , the configuration $x$ will
%be usually given by the context (though usually not in the whole
%$\zv$!) and so we will often call the admissible system $\Cal D$ \
%small not specifying the configuration $y$ which is uniquely
%determined only in the vicinity of $\Cal D$. Let us make the agreement
%that $(x,\Cal D )$ will be called small if it is small for {\it some}
%extension of the given $x$ to the whole $\zv$.
%The notion of an {\it extremally small\/} admissible system $\Cal D$ in some
%finite volume
%--with nonspecified $x$-- will be
% defined as above; the requirement will be that $\Cal D$ is
% extremally small for {\it some} configuration $x$.
% \newline
%{\bf 4. }
A typical example of an extremally small system $\Cal D $ is
a collection of the type
$\Cal D =\gb_{\ext}\&\{\gb_i : \gb_i \to\gb_{ext}\}$ where
the contours $\gb_i$ are {\it not\/} small
but the whole system $\Cal D $ is small.
The case $\{\gb_i\}=\emptyset$ is the most common one, of
course.
\newline
{\bf 4.}
One should again 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 have even 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 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} A configuration $(x, \Cal D)$ of a mixed model whose all strictly
external subsystems are small contains at least one extremally small subsystem.
\endproclaim
\demo {Proof} Consider the decomposition of $\Cal D$ into minimal strictly
external (by a strict
externality of $\gb$ in $\Cal D$ we mean that
$\vv \cap \supp(\Cal D \setminus \gb =\emptyset$) subsystems.
Denote them as
$\{\gb_i \ ; i = 1,2,\dots \}$. Take the smallest, in the geometrical
sense of $\prec$ ,
external subsystem, say $\gb_1$,
of this configuration. Now, if the collection of extremally small
subsystems (equivalently, by induction, the collection
of small subsystems)
$\gb' \subsetneqq \gb_1$ of $\gb_1$ would be empty then
$\gb_1$ itself would be extremally small!
\enddemo
The importance of the notion of a small resp. extremally small
subsystem stems from the fact that extremally small subsystems
provide practically the only
relevant {\it example\/} of recolorable subsystems.
\footnote{One
can construct, of course, examples
of recolorable but not extremally small systems.
They correspond, however, just to some marginal cases not covered
by the particular choice of the constants $\tau,\tau',\tau(\gb)$
in the definition of a small contour $\gb$.} The following
result gives such a statement. It is a crucial step (together
with Theorem 5 above) in the proof of the forthcoming
Main Theorem.
\proclaim{Theorem 6} If $\tau' = \frac {\tau}{ 12 \nu} $ then any
extremally small
subsystem is recolorable.
\endproclaim
(We mean the values from (2.17) and (3.34')).
Theorem 6 will imply that after the completion of the recoloring procedures
of Theorem 5 (applied to the original P.S. model), {\it no\/} small
contours or subsystems will be left in the final mixed model.
This leads to the Main Theorem, see section 8.
%A possible choice of these constants is
% $ \check \tau = \tau' = {\tau \over 2} $
%Then
%$\tau'$ in (3.34) can be taken something like $\tau \over 4 $
%for $\tau$ sufficiently large.
\endremark
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head {6. The proof of Theorem 6 }
\endhead
Say that $\gb$ is {\it tight\/} if any its removable subcollection $\Cal D$
%(which can {\it not\/} be small by the very definition of
%extremal smallness )
satisfies the bound
$$ \dist (\square (\Cal D) , \gb \setminus \Cal D ) \leq
\diam \square (\Cal D).\tag 3.52
$$
(Recall that we take the $l_{\infty}$ norm everyhere.)
We will show that the proof of Theorem 6 can be reduced to the
case of extremally small tight systems $\gb$. Then
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 $\gb$!)
and Theorem 6 will be simply 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 the more
characteristic and important for the proof. The quantities $A(\gb)$ (more
specifically, the quantities $A^*_{\text{full}}(\square)$)
will then play an important role in this reduction to tight systems.
\definition { The skeleton of $\Cal D$}
\enddefinition
Consider the following
auxiliary construction in any volume $\Lambda$, for any
(locally) stratified configuration $y$ on $\Lambda$.
We will actually use it below with the special
choice $\Lambda =
V_{\updownarrow}(\Cal D)\setminus \supp \Cal D $ (for a contour or
admissible
system $\Cal D$) and with $y$ being the local colour induced by
$\Cal D$ on $V(\Cal D)$.
\definition{Definition}
Given a stratified configuration $y$ and a cube $\square$ say that $\square$
is {\it minimal nonsmall\/} if $A^*(\square , y) > \tau' ( \diam \square)$
(see (3.49)) and no smaller cube which is at the same time a subcube
of $\square$ satisfies
such a condition. \footnote{Sometimes, for ``stable'' $y$ (defined below in
Main Theorem) such a cube will not exist; however this is
not the case of a
typical nontrivial situation below.}
Given a volume $\Lambda$, let us find
some smallest possible (in $\prec$) minimal nonsmall cube
$\square\subset \Lambda$ (if there is one).
Take all the adjacent (having distance $1$ to $\square$) cubes
$\square'\subset \Lambda$ 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 go outside of $\Lambda$ are excluded!)
of cubes inside $\Lambda$.
Construct also other possible
partial layers {\it not touching\/} 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''
is that (e.g.) the vertical distance between any two
adjacent layers is
bigger than the {\it logarithm\/} of the thickness
of both layers. Why we require this will be expained below.
\footnote{ The reason is to keep $\Af$ 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 $\Lambda$.
The same construction can be defined, in analogous way, in any
(generally nonstandard) volume with a given {\it locally stratified\/}
configuration. In particular, a {\it skeleton\/} of the
interior $V(\Cal D) \setminus \supp \Cal D$ of any
{\it extremally small system\/} $\Cal D$ can be thus constructed.
\enddefinition
\remark{Note}
The fact that skeleton has a ``smallest possible
grain'' is maybe slightly superfluous here but
it will surely be useful not only below but also in some other,
more detailed estimates (like those
used in the study of the completeness of
the phase picture constructed by Main Theorem).
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 interior (and therefore nonsmall)
$\gb \subset \Cal D$ is {\it at least as big\/} as the
size of the nearest neighboring cube from the skeleton (if there is one).
%It seems that for a mere proof of Main Theorem, one apparently does not
%need such a detailed knowledge about the skeleton.
% ( We are
%here preparing our treatment also for the future, more detailed
%)
\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(\gb) \setminus \supp \gb $
which is composed of nonsmall {\it cubes\/}. These
cubes are ``densely packed'' as formulated above.
\definition{Rearrangement of the energy of $\Cal D$}
\enddefinition
Let us define the following ``rearrangement''
of the energy of a given extremally small configuration
$\Cal D$:
Imagine that the cubes of the skeleton are just some new ``contours''.
%-- whose possible subordination to contours of $\gb$ is given in
%the natural way\footnote{A cube of the skeleton intersecting
%the volume $V(\gb)$ of some contour $\gb$
%of $\Cal D$ is subordinated to it i.e.
%we have an arrow $\to$ from such a cube to the given contour $\gb$.}.
The idea is to show that such an ``enrichened'' system $\Cal D^*$
of ``contours''
is {\it tight\/} in the sense (3.52),
and the value of its contour functional $F(\Cal D^*)$
(see (3.54) 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) for the
case when $\Cal D =\gb$ 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 $\square = \gb$ we put (just to unify the notations
in the formula (3.54) below!)
$$ | \supp \gb| = \diam \square . \tag 3.53 $$
With this notation, using the Peierls condition (2.17) and the
definition (3.22) of $F(\Cal D)$ we can prove
(3.34') by showing the following inequalities:
Recall that
$$A^*(\square,x) \geq \tau' \diam \square \tag 3.49'$$
holds for any
cube of the skeleton. See (3.49); $x$ denotes
here the configuration induced by $\Cal D$ on $\square$ (and extended somehow
to the whole $\zv$).
Now we have the following relation between the contour functionals
of the original extremally small
system $\Cal D$ and the (``enrichened'', by cubes of the skeleton)
system $\Cal D^*$:
\proclaim{Proposition} We have the relation
$$ \tau |\supp \Cal D| - A(\Cal D) \geq \tau' |\supp \Cal D| - \Af(\Cal D)
\geq
\tau' |\supp \Cal D^*| - \Af(\Cal D^*) .
\tag 3.54 $$
\endproclaim
To prove this, notice that the first inequality just informs
us about the approximation
of $A$ by $\Af$ (compare (3.42),(3.45) and (3.47))
while the second inequality will be shown now to
be a {\it consequence of the very definition of the skeleton\/}.
Write (3.49') as
$$ \tau' |\supp \gb| - \Af(\gb) \leq 0 \tag 3.49'' $$
for any new ``contour'' $\gb =\square$ of the skeleton of $\Cal D$.
\footnote{What follows will be just a suitable
play with the quantities
$\Af|\square|$ 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 $\square$''.}
The idea of the proof of (3.54) is
that the terms $F(\Cal D)$ resp.
more precisely
$\tau'|\supp \Cal D| -\Af(\Cal D)$ are essentially {\it additive\/}
(as functions
of the components $\gb$ of $\Cal D$). The
additivity of the functions of the type $E(\Cal D) = \sum E(\gb)$
(where the sum is over all contours of $\Cal D$)
is of course trivial.
Concerning the approximate additivity of the function $\Af$
we have the following auxiliary result.
\footnote{The quantity $A(V)$ is of course {\it exactly\/} additive for
disjoint volumes $V$ but it would have some other,
more severe disadvantages (than $\Af$) when the ``surface tension'' along
the vertical sides of the cubes would be discussed.}
\proclaim{Lemma}
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 horizontal distance between any two cubes from $\Cal S$ 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 $\Cal C$)
$$ \Af(\Cal D^*) = \Af(\Cal C) + \sum_{\square \in \Cal S}
\Af(\square S) + D(\Cal D^*) \tag 3.55
$$
where the correction term $D$ satisfies the bound (with some
large $\tilde \tau$)
$$ |D(\Cal D^*)| \leq \sum_{\square \in \Cal S}( \diam \square)^{-\tilde
\tau} . \tag 3.55D$$ \endproclaim
\demo{Proof} Consider two such collections $\Cal D^*$
which differ just by {\it one
cube\/} $\square$ \ i.e. let \ $\Cal D^{**} = \Cal D^* \& \square $.
Assume that the cube $\square$ is no greater than any cube of $\Cal D^*$.
It is now sufficient to prove the bound
$$ |D| = |\Af(\Cal D^{**}) -\Af(\Cal D^*) -\Af(\square)|
\leq (\diam \square)^{- \tilde \tau}. \tag 3.55D' $$
Notice that (3.43) can be written also as the sum over intervals $I$
$$ f_t(y) = e_t(y) - \sum_{I \subset \zet\ : \ t \in I} k_I \tag
3.43I$$
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
(3.55I) 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
$$ A^*_{\text{ignore}}(\Cal D^{**}) - A^*_{\text{ignore}}(\Cal D^*)
- A^*_{\text{ignore}}(\square) = 0 \tag 3.55I$$
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 (3.55I), and therefore also (3.55D').
\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
\footnote{ Notice that our use of the {\it squares\/} in the
definition of smallness is rather important technically. Namely,
our method of the proof relies quite heavily here
on the fact that the enriched
(by cubes of the skeleton)
system $\Cal D^*$ of any small $\Cal D$ is again a small system!}
smallness of the cube $\square(\Cal D^*)$ !)
$$ \tau'|\supp \Cal D^*| -\Af(\square(\gb^*))
\geq \tau' (|\supp \Cal D^*| - \diam \square(\Cal D^*))
\tag 3.56 $$
which is surely {\it greater\/} (notice that
$ \diam \square(\Cal D^*)= \diam \square(\Cal D)$!)
than, say, ${\tau\over2} |\supp \Cal D^*|$.
Thus it suffices to show now that the quantity $ {\tau \over 2}
|\supp \gb^*|$ (and, therefore, also the right hand side of
(3.57))
is greater than, say, \ $ {\tau\over 12\nu} \conn \gb$.
We will prove this by
proving the following result (Theorem 7).
First generalize the notion of a tight system $\gb$ to any
subset of $\zv$:
\definition{Tight sets}
\enddefinition
Say that $S \subset T$ is isolated in $T$ ($ T \subset \zv $)
if
$$ \dist(\square(V_{\updownarrow}(S)), T \setminus S) \geq \diam
\square(V_{\updownarrow}(S)) .\tag 3.52' $$
Say that $T$ is {\it tight\/} if there are no isolated subsets of $T$.
\remark{Notes}
1. The choice of $ \square(V_{\updownarrow}(S))$ above is of course
motivated by our definition of $\square(\gb)$ and our emphasis
on the notion of a {\it strict\/} interiority and diluteness.
``Topologically'', there is not much difference between
the choice above or the choice of the cube $\square(S)$.
\newline
2. Notice that the system $\supp \gb^*$ is already tight
because the definition of a skeleton of $\gb$
gives no room for subsets of the type (3.52) for the
enriched set $T = \supp \gb^* $ !
\endremark
\proclaim {Theorem 7} If $T$ is tight then
$$ \conn T \leq 6\nu |T| . \tag 3.57$$
\endproclaim
Let us start the proof of Theorem 7 by the simple observation (used
in Lemma 1, section 7 below) that any contour of $\gb^*$ can be assumed to have a
a cardinality at least $1024 = 2^{10}$.
Really, it is rather straightforward to see the
validity of a bound
$$ \conn \Cal D \leq 9 |\supp \Cal D| \tag 3.57'$$
for any interior tight subsystem $\Cal D $ of $\gb$ whose cardinality is
$d \leq 1024$.
To check this bound remember that our contours
(supercontours in the sense of Chapter 2; this is the second
moment -- after the proof of Theorem 5 -- where we profit
from their properties) are such that
$\gb \rightarrow \gb'$ implies that the cardinality
of $\supp \gb'$ is at least twice bigger than, say
the cardinality of $\supp \gb$.
Thus, the longest branch of the forest $\Cal D$ has at most
$9$ sites.
\newline
Now take any tree of $\Cal D$. Retain the notation
$\Cal D$ for it, and for any subtree $\Cal E \subset \Cal D$
construct a suitable
commensurate path connecting $\Cal E$
to some superordinated contour of the remainder of $\Cal D $.
The length of any such path can be surely chosen smaller than, say,
the cardinality of the support of $\Cal E$:\newline
Namely, consider the projection to $\zet^{\nu -1}$ : then all
the projections of contours of $\Cal D$ are connected.
Construct first such a connecting a path in the
projection to $\zet^{\nu -1}$, with ``horizontal'' steps of length $1$
(surely, less than $d =\diam \Cal E$ steps are needed), and
then add suitable vertical components to the already constructed
horizontal steps to keep the successive steps commensurate and to guarantee
that the path (not only its projection) really
starts in $\Cal E$ and ends in $\Cal D \setminus \Cal E$.
The vertical distance to be overcomed is also smaller
than $d =\diam \Cal E$ and clearly, a suitable choice of $d$
commensurate vertical steps (which can be understood
as vertical components of the horizontal unit steps
constructed above) overcoming the given
vertical distance is also possible.
Now, when comparing the total length $\sum l(P)$
of the these paths $P= P(\Cal E)$, connecting any interior subtree $\Cal E
\subset \Cal D$
to the remainder of the corresponding tree of $\Cal D $,
with the sum of the cardinalities $|\Cal E|$ of the subtrees
(which surely have at least as many points as
is the length of the path $P(\Cal E)$!)
we see that $\sum l(P(\Cal E)) < 9 |\supp \Cal D|$
because each point of $\Cal D$ was used at most 9 times in
the above consideration.
\newline
(This statement is actually
some ``weaker'' analogy of Theorem 7 for systems having
contours with a cardinality less than $1024$.)
Moreover, by the same argument one can connect any such ``small sized''
$\Cal D$ to the superordinated tree of the remainder of $\gb$
by a path containing no more than $|\supp \Cal D|$ of additional steps.
The conclusion is that $10 |\supp\Cal D|$ is surely the upper bound
for the number of points needed to make
each such $\Cal D$ commensurately connected and connected also to remainder
$\gb \setminus \Cal D$. Thus we can really
restrict ourselves to
the case when the smallest interior components of $\gb$
have a cardinality at least $1024$. %: recall
%that the supports of $\gb$ are only ``half connected''
%by Theorem 2 and proposition of section 7, part II;
%so $12 |\supp\Cal D|$ (11 = 9 + 2) is the final upper bound
%for the number of points needed to make
%each such $\Cal D$ commensurately connected and connected also to
%$\gb \setminus \Cal D$.
\remark{Note}
Notice that
after the removal of any such ``small sized'' subtree $\Cal D$, the remainder
is still extremally small (if the original admissible system was).
\endremark
Thus it suffices to prove the desired inequality
$$ \conn \gb^* \leq 6 \nu|\supp \gb^*|\tag 3.58 $$
at the assumption that all the contours of $\gb^*$
have already a (halfconnected: recall Theorem 2 and
Proposition in section 2.7 )
support having a cardinality
at least $1024$. If we assume contours to be moreover
{\it connected\/} then it suffices to show (in slightly more
general setting) that the inequality
$$ \conn S \leq 3 \nu |S| \tag 3.59 $$
holds for any $S \subset \zv$
at the assumption that the components of $S$
have cardinality at least $1024$.
%Now, we first generalize the notion of a tight system $\gb$ to any
%subsets of $\zv$ :
% Say that $S \subset T, T \subset \zv $ is isolated in $T$
%if
%$$ \dist(\square(S), T \setminus S) \geq \diam \square(S) .\tag 3.52' $$
%Say that $T$ is {\it tight\/} if there are no isolated subsets of $T$.
%
%Clearly, the system $\supp \gb^*$ {\it is\/} tight
%(the definition of a skeleton
%gives no room for subsets of the type (3.52) for $T = \supp \gb^* $ ! )
%\proclaim {Theorem 7} If $T$ is tight and all the
%connected components of $T$ have a cardinality
%at least $1024$ then
% $$ \conn T \leq 6\nu |T| . \tag 3.58$$
%\endproclaim
Apparently, the proof of such a geometrical statement will finish
also the proof of Theorem 6. Namely, then we can conclude the
arguments of (3.54) and (3.55) as follows:
$$ \tau |\supp \Cal D| - A(\Cal D) \geq \tau' |\supp \Cal D^*| - \Af(\Cal D^*)
\geq
\tau'|\supp \Cal D^*| - \tau' \diam \square(\Cal D^*) $$
i.e. $$ \tau |\supp \Cal D| - A(\Cal D)
\geq {\tau \over 2} |\supp \Cal D^*|
\geq {\tau \over 12\nu} \conn \Cal D . \tag 3.60 $$
\definition {Proof of (3.59). Commensurately connected collections of cubes}
\enddefinition
It will be useful to {\it reformulate\/} first the notion of a commensurately
connected graph in an alternate language -- based on the employment
of {\it cubes\/} from $\zv$ instead of the 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 1}
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 2}
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 :
$$ a' = a + v_1 + v_2 + v_3 \tag 3.62 $$
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 of the type above is actually repeated in the
direction
of any coordinate axis for $\nu \geq 1$)
is commensurately connected.
\enddemo
\proclaim{Corollary}
Let $S \subset \zv $. Let $\conn_{\square}$ denote the cardinality
of a smallest commensurately connected collection of cubes
%$\zv$ containing $S$
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 $1024$.
\definition{ Second covering cube }
\enddefinition
Define now 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{ Definition }
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 $\square$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 yet.
\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 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 now 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 statement :
\proclaim {Lemma} 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
This is easily seen (it suffices to consider the case of
the dimension $\nu = 1$!) from the inequality ($d$ denotes the distance
between two cubes of diameters $1$ ($\square'$) resp. $x$ ($\square''$))
$$
d \leq 1 \ \& \ 1/2 \leq x \leq 2
\Rightarrow (1 + x + d)^{1 \over 2} \leq 1 + x^{1 \over 2}.
\tag 3.66$$
\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}
Theorem 6 is a stronger and better replacement for the ``Main Lemma''
of [Z].One could use its 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 contours''
such that $a|V(\gb_i) > \tau |\supp \gb_i|$ and if we denote by
$\ext = \Lambda \setminus \cup V(\gb_i)$ then
$$ a |\ext| > \tau \Conn(\Lambda,\{\gb_i\}) \tag 3.71 $$
where the integer on the right hand side denotes the cardinality
of a smallest possible set whose union with $\Lambda^c$ and
all $\supp \gb_i$ is connected.
With this lemma, one can rewrite the ordinary P.S. theory
in a way analogous to that used here {\it without\/} an explicit construction
of the
contour models. See the lectures notes \cite{ZRO}.
\endremark
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 8. The Metastable Model. The Main Theorem.
\endhead
Theorems 5 and 6 imply that the process of recoloring does not stop
up to the very moment
when there are {\it no\/} recolorable removable subsystems available
in the
final mixed model. This final mixed model will be called the
(totally)
{\it expanded Pirogov -- Sinai model\/}, corresponding to the original abstract
Pirogov -- Sinai model with the hamiltonian (2.16) satisfying (2.17).
Thus we have the following result.
\proclaim{Corollary}
In the expanded model, only those
configurations of contours remain which contain
no strictly interior recolorable subsystems. In other words,
which are represented (Theorem 1)
by a graph
(without loops) whose any complete subgraph (complete in the relation
$\gb \to \gb'$ : if the end of the arrow is in the subgraph then
the whole arrow is also in the subgraph) which is moreover strictly
interior is
{\it nonrecolorable\/} (therefore also nonsmall). \endproclaim
\definition{Definition}
Configurations of contours of the type above will be called
{\it redundant\/}.
\enddefinition
\remark{Note}
An equivalent characterization of redundancy in terms
of the related notion of metastability will be given below.
Notice that the extremal smallness of $\gb$ means
just the {\it smallness of $\gb$\/}
\& {\it redundancy of all strictly interior subsystems of $\gb$ \/}.
Speaking about the nonsmallness of some $\gb$ in Corollary
above one has
in mind the nonsmallness of $\gb$ in the
``provisional'' mixed model, constructed up to the moment
when the extremal smallness of $\gb$
is checked before applying
Theorem 5. In this
provisional mixed model, there are already {\it no\/}
extremally small systems which would be ``geometrically smaller
than $\gb$''\ (in $\prec$).
\newline
By the way, nonsmallness of a contour (or admissible system) $\gb$
in the
final, fully expanded model and in the provisional one
would mean almost the same.
More precisely we note that for cubes $\square$ of a comparable (or smaller)
size as
$\gb$ the quantities $A^*(\square)$ and $A^*(\square)$ are practically
the same for both the provisional model (expanded ``up to the size of $\gb$'')
and the final expanded model. (The quantities $A(V)$ are {\it exactly\/}
the same for volumes $V \subset V(\gb)$, in both models.)
Namely,
the clusters $T$ having a different value
in both considered mixed models (the provisional one and the final one)
have a size at least $\diam \gb$
and the difference between the corresponding values of
$A(\gb)$ is thus 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) below)
and also in (3.90).
% Notice that the arguments leading to the proof
%of Theorem 6 are sufficiently ``robust'' with respect to possible such
%modifications in the
%definition of the smallness of $\gb$.
\endremark
\definition{The metastable models}
Say that a subcollection $\Cal D$ of an admissible system $\tilde \Cal D$
is an {\it external\/} one if $\tilde \Cal D \setminus \Cal D$
can be decomposed into disjoint collections
$\Cal D'$, $\gb_1$, $\gb_2$,\dots such that
$(V(\Cal D)\cup_i V(\gb_i)) \cap V(\Cal D') = \emptyset$
\ and moreover $ \gb_i \to \Cal D$ for any $ i\geq 1$.
\newline
A configuration $x = (x_{\text{best}},\gb) $ which is $y$ -- diluted
(i.e. equal to
$y \in \Cal S$ outside of some set having
finite components which are standard sets) will be called
{\it $y$--th metastable\/} (shortly, metastable)
if no redundant external subsystem $\Cal D$
of $\gb$
exists. The restriction of the
original abstract Pirogov -- Sinai model (with the hamiltonian (2.16))
to all $y$--th metastable configurations
will be called the $y$--th metastable model.
If the abstract Pirogov -- Sinai model was constructed as a representation
of an original model (2.2) then we define the metastable $y$-- th 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 $x =(x_{\text{best}},\gb)$ in terms
of their contours are $y$--th metastable in the sense above.
Now we can also say that an admissible system $\Cal D$
is redundant if and only if it has {\it no removable metastable subsystems\/}.
\enddefinition
\definition{Notation}
Recall the quantities $f_t(y)$, $y\in\Cal S$
-- see (3.43) -- which were constructed for any
mixed model.
Consider now these quantities for the case of the totally expanded model
whose construction was just finished.
For the expanded model corresponding to the original
Pirogov -- Sinai abstract model, the quantity $f_t(y)$ will be denoted as
$$
f_t(y) \ = \ h_t(y) . \tag 3.72
$$
\enddefinition
Now we are able to formulate the basic result of the paper:
\definition{Stable elements of $\Cal S$}
A stratified configuration $y\in\Cal S$ for which there is
{\it no\/} redundant contour or admissible system $\gb$ such that
$(x_{\gb})^{\ext}= y$ will be
called {\it stable\/}.
\newline In other words, $y$ is stable if the collection of
configurations $ x\neq y$ having the value $y$ outside $\Lambda$
is empty in the fully expanded model.
\enddefinition
\proclaim{Main Theorem} Consider an abstract Pirogov -- Sinai model
defined
by (2.16) and (2.17), with $\tau$ sufficiently large.
Then the quantities $h_t(y)$ constructed by (3.72)
are the free energies of the
corresponding $y$-- th metastable models. The
configurations $y \in \es$ which are stable correspond
precisely to those configurations from $\es$ which are the
ground states of the quantity
$$
\sum_{t'\in [\,t\,]} h_{t'}(y) \tag 3.74
$$
where $[\,t\,]$ denotes the collection of all \ $ t'= t +
(0,0,\dots,0,t_{\nu}); t_{\nu} \in \zet $. The ``ground
state'' is understood in the sense that we always have
$$
\sum_{t'\in[\,t\,]}(h_{t'}(\tilde y)-h_{t'}(y) ) \geq 0
\tag 3.75
$$
if $\tilde y \in \es$ differs from $y$ on a layer of a finite
width. For any stable $y $ there exists a probability measure $P^y$
on the configuration space of the given abstract \ps model
whose almost all configurations are $y$-th metastable and moreover, the
conditioned probabilities $P^q_{\Lambda}$ of $P^y$,
being taken with respect to all configurations which are
$y$ -- diluted resp. strongly diluted in $\Lambda$
correspond to the diluted resp. strictly diluted
ensembles (2.18).
\endproclaim
\proclaim{Corollary} If the considered abstract Pirogov -- Sinai model
represents a ``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 support
can be identified as a suitable
subset of the collection $\ex_{\text{meta}}^q$
of all $y$-th metastable configurations.
\endproclaim
By a support of a probability measure we mean
a Borel (more precisely countably compact) set having measure $1$.
\remark{Notes}
{\bf 0. } Clearly, the families of configurations
$\ex_{\text{meta}}^y$ are mutually disjoint for different $y \in \es$.
We do not study here in much detail the structure of a typical
configuration (of a $q$ -- th Gibbs state) here. See, however,
the final section 9 for some information.
(This problem deserves a more full treatment. However, it seems
reasonable to do this in connection with some future
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 $).% and postpone the
%corresponding discussion to a forthcoming paper.
It can be done similarly as in \cite{Z}.
See also some comments in 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 ``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''(the components of $\vv$
where $\gb $ is the collection of all contours of the
the considered configuration
$x$ ; this covers 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 expansions (3.43)
(within a given precision; of course this is in full
a horribly complicated sum -- but its terms are converging
{\it very\/} quickly, indeed), 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 ``$M$--expanded''
model ($M$ is some square, for example)
where only those extremally small subsystems whose size
does {\it not\/}
exceed the size of $M$ are already recolored.
In fact, even for squares $M$ quite small some
useful approximations can be found, often enabling already 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 Blume Capel models, even the smallest size 2$\times$2 of the square $M$
can be useful -- see \cite{BS}. Namely, considering only
first two or three terms in (3.43) a correct conclusion
about the stability of $0$ on one side and $+/-$ on the
other side can be made.)
\newline
{\bf 4.}
In fact, in finite volumes $\Lambda $ there is no noticeable
difference between the behaviour of the stable $y$ --th
phase and another $\tilde y$ --th phase if the quantity
$$
a=\sum_{t\in[\,t\,]}h_{t}(\tilde y)-h_{t}(y)
$$
is such that, say, $a^{-1} >(\diam \Lambda)^{\nu -1} $. Quite straightforward
estimate of the quantities
$$
A(\square) \leq p(V(\square)) \leq
\tau \diam \square \leq \tau (\diam \Lambda)^{\nu -1} p(V(\square))
\tag 3.76
$$
where $p$ denotes the orthogonal projection on $\zw\subset
\zv$ shows that 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 if a is very small; namely the difference between
various $h^y$ is only of the order $\exp (- F(\gb))$ ) if
$e^y$ are the same and $\gb$ is the smallest contour
``which makes the difference'' between two different $y$.)
\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 (3.75).
%$$
% \sum_{t\in [t]}h_{t}(y)-h_{t}(\tilde y)\geq 0\, .
%$$
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 is not a paper
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 were already Theorems 5 and 6 above -- which guarantee
the very existence of the fully expanded model and therefore the possibility
of the very formulation
(based on the existence of the
quantities $h_t$) of our result.
Noticing this, one has to add now only a few additional
observations :
1) If $y$ satisfies (3.75) then for any admissible system $\gb$
such that $(x_{\gb})^{\ext}=y$ we have, from the very
definition of the quantity $A^*(\square)$, $\square =\square(\gb)$
(see (3.37) and the commentary below )
the inequality
$$ A^*(\square) < \varepsilon \diam \square. \tag 3.80 $$
Really, one could take even {\it zero\/} on the right hand side
if the appropriate partially expanded model (namely
the mixed model studied at the moment
when the smallness of $\gb$ was discussed)% but {\it not} established
could be taken here.
However, we have
a different mixed model now
(at the end of the recoloring procedures)
-- the fully expanded one -- and
we have to use (3.36)) to notice that
$$|A^*_{\text{partially expanded}}(\square) - A^*_{\text{expanded
}} (\square)| \leq \varepsilon' |\diam \square| \tag 3.81$$
where $\varepsilon'$ is {\it very\/} small, of the
size $\varepsilon^{\diam \square}$ !
Thus we see, that the smallness of any $\gb$
with the external colour $y$ satisfying (3.75)
is almost a tautology
and therefore, under such boundary condition $y$, there is {\it no\/}
difference between the original \ps model and the metastable one.
Moreover one has quickly converging expansions
of partition functions with the boundary condition $y$
in any volume, and this implies the validity of the properties of the
``$y$ -- th Gibbs state'' stated in the Main Theorem.
One could prove also the exponential decay of correlations in
any Gibbs state thus constructed. See the last section 9 below
for some relations (namely (3.90)) proving (or, at
least, preparing a ground
for the proof) of these facts.
2) On the other hand, if
$y$ is {\it not\/} a ground state then there is some
$\tilde y$ differing from $y$ on some layer $L$ of a finite
width -- say $d$ -- such that the vertical sum is
$$
\sum_{t'\in[t]}h_{t'}(y)-h_{t'}(\tilde y) \geq \delta
\tag 3.82 $$
for 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 (which has the form of a ``desk'')
$$
\Lambda = \{t\in\tilde L, \hat t \in B\}
$$
where $\tilde L$ is another (thicker) layer containing $L$
``sufficiently inside''.
Take the configuration $x=y$ outside $\Lambda$,
$x=\tilde y$ inside $\Lambda$. Then, if we compute the
quantity $A^*(\Lambda)$ for the volume
$$
\Lambda = (\partial B\times\zet)\cap L
$$
we have, according to (3.45), the bound from below:
$$
A^*(\gb) = \sum_{t\in\Lambda}(h_t(y)-h_t(\tilde y))
\geq
\delta |B| -\varepsilon |\supp \gb|>>
\tau d|\partial B| >> \tau|\diam \Lambda|
\tag 3.84 $$
according to (3.49) which shows (compute $A^*(\square)$
for $\square \supset \Lambda$ : this is even bigger than
$A^*(\gb)$ !)
that the ``contour $\gb$
encircling the cylinder $\Lambda$''
is {\it not\/} a small contour!
Strictly speaking, this argument requires some commentary --
because we do {\it not\/} know that such a $\gb$ is a contour of the model.
This can be done precise simply by {\it allowing also contours
with the weight $+\infty$ \/} \ in our abstract Pirogov -- Sinai model;
such an assumption
does not change its ``physical'' properties but allows to
work with a richer family of contours (which is quite indispensable here,
as we see in (3.84)).
\enddemo
\remark{Note} This is only one of the arguments why it is generally advisable
to work with an abstract Pirogov -- Sinai model instead of the
original spin model.
\endremark
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vskip1mm
\head 9. Properties of Typical Configurations. Concluding Notes
\endhead
We are still in a situation of an abstract Pirogov -- Sinai model.
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 $x =(x^{\Lambda}_{\text{best}}, \gb) $
% of $ X^y_{\text{meta}}$
such that
$\gb$ is strictly diluted in $\Lambda$
i.e. $\vv \subset \Lambda$.
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 the Gibbs density
$\exp(-H(x_{\Lambda}|y))$ (and $H(x_{\Lambda}|y)$ being given by (2.16)).
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 a suitable limit over $\Lambda \to \zv$
$$ P^y_{\text{meta}}(\Cal E )
= \lim_{\Lambda \to \zv} P_{\text{meta},\Lambda}^y(\Cal E)
\tag 3.85 $$
exists for a sufficiently rich collection of events $\Cal E$
(``sufficiently rich'' would mean, in the 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 as below in (3.86))
and gives, for $y$ {\it stable\/}, a Gibbs measure $P^y$ on the whole $\ex$.
Let us show for example that the limit
$$ P^y(\Cal E (t)) = \lim_{\Lambda \to \zv}
P_{\Lambda}^{y}(\Cal E(t)) \tag 3.86
$$
exists for any (even nonstable) $y$ and
for any $t \in \zv$, for the event $\Cal E(t)$ defined as follows:
``$t \notin \vv$, where $\gb$ denotes the
collection of all contours of $x$'' i.e. for
the event ``$t$ is stricly exterior point of the given configuration $x$''.
We have obviously the formula
$$P_{\Lambda}^y(\Cal E(t)) =
(Z_{\updownarrow}^y(\Lambda))^{-1} Z_{\updownarrow}^y(\Lambda \setminus t)
\exp(-e_t). \tag 3.87 $$
We can expand, by (3.0)
(in the fully expanded
model)
both the partition functions
on the right hand side of (3.87). The possibility of such an expansion
follows from the very notion of stability of $y$! (resp. from the definition
of the metastable ensemble, if $y$ is unstable
and we compute the probability $P^y_{\Lambda,\text{meta}}$).
We have, for $\dist(\Lambda, t) \to \infty $, the following expression:
(Analogous, slightly more cumbersome expressions
obtained in any finite volume $\Lambda$ are omitted here.)
$$
\log P^y(\Cal E(t)) = \sum (k_T^{\text{ext}} -
k_T) \tag 3.88 $$
where the quantities $k_T$ resp.
$k_T^{\text{ext}}$ correspond
to the expansion of $Z^y_{\updownarrow}(\Lambda)$) resp.
$Z^y_{\updownarrow}(\Lambda\setminus t)$) and
the sum is over those (quickly decaying by (3.1)!) quantities
$k_T$ resp. $k_T^{\text{ext}}$ only which
{\it touch\/} $t$ (contain or have a distance at most $1$).
Clearly, for sufficiently small $\varepsilon$ we have from
(3.1) and (3.88) the approximate relation
$$ 1 - P^y(\Cal E(t)) \ \asymp \ \varepsilon . \tag 3.89 $$
This suggests that the ``islands'' of a typical configuration $(x,\gb)$
(interpreted here as connected components of $\vv$,
but see below
for a more elaborate notion of an ``island'') are really typically
``small and rare''
because they do not typically intersect a given (arbitrarily chosen)
point $t \in \zv$.
To make this intuitive description of a typical configuration
(which is quite characteristic for the Pirogov -- Sinai theory and
the phase picture this theory gives)
more detailed, define below the ``islands''
of a given configuration $(x,\gb)$ in a different, more detailed
way grasping also some important features (namely
the appearance of redundant contours) of the regime
appearing {\it inside $\vv$\/} and thus allowing also precise expansion formulas
for the
event ``a given island appears'':
\definition{Definition}
An admissible subsystem $\gb \subset \Cal D$ of a configuration $(x,\Cal D)$
is called an {\it island\/} if there is a strictly external (in $\Cal D$)
superset
$\gb' \supset \gb$ such that $V(\gb' \setminus \gb) \cap
\gb = \emptyset$ and $\gb' \setminus \gb$ contains no
redundant subsystems. (In other words, a successive recoloring ``around $\gb$''
can be applied, deleting all the elements
of $\gb' \setminus \gb$. The case $\gb' =\gb$ is typical, of course,
and $\gb$ is most commonly either a single contour or a collection of
one external contour and some interior redundant ones.)
%$\gb \subset \Cal D, \vv \cap V_{\updownarrow}(\Cal D \setminus \gb)
%= \emptyset$ and moreover, no redundant $\gb' \subset \Cal D$ exists such that
%$\gb \cup \gb'$ is admisssible.
Denote by $P^y[\gb]$ the probability (in $P^y$) of the
event ``$\gb$ is an island of a given configuration''. \enddefinition
\proclaim{Proposition}
$$ P^y[\gb] = \exp (-F_{\infty}(\gb)) \exp(\sum_T (k_T -k_T^{\ext}))
\tag 3.90 $$
where $F_{\infty}$ is defined as in (3.22),(3.21) but with
respect to the {\it fully expanded model\/}
(not the temporary one, used in the moment when $\gb$ was recolored!)
and the sum is over those $T$ only which touch $\supp \gb$. The quantities
$k_T$ resp. $k_T^{\ext}$ corrrespond to the quantities
$k_T(\gamma\delta)$ resp. $k_T(\delta)$ in (3.25).
\endproclaim
\remark{Note}
1. Notice that $F_{\infty}$ is a horizontally translation
invariant quantity.\newline
2. Notice that we do not formulate here the probabilities
of the events depending also on the
{\it interior\/} of $\gb$.
Having determined all possible $P^y_{\gb}$ this is already (in the
strictly diluted ensemble) a straightforward
task, using the properties of
conditioned Gibbs distributions in finite volumes.
\endremark
\demo{Proof}
This is just an application of (3.25), for $\Cal D = \emptyset$.
It is not {\it exactly\/} the same argument as the one which was used
in section 3 for
the recoloring of $\gb$. Namely, a (slightly) different mixed model
is used here, also with clusters ``bigger than $\gb$'';
however the difference between $F(\gb)$ and $F_{\infty}(\gb)$
is extremely small, of the order $ \varepsilon^{\diam \gb}$.
\enddemo
\proclaim{Corollary}
The probability $P^y[\gb]$ of an island $\gb$ satisfies an estimate,
with $\varepsilon = \exp(-\tau')$ where $\tau'$ is something like
$\frac{\tau}{ 13\nu}$
$$ P^y[\gb] \leq \varepsilon^{\conn \gb}\ . \tag 3.92 $$
The probability of the event ``there is an island around $t$
having a diameter $\leq d$'' can be estimated
as $\text{const}\ \varepsilon^d$.
The mean relative area, occupied by the islands $\gb$ resp. by the
``interiors'' $\vv$ of a typical configuration
is smaller than some $\varepsilon' =
\varepsilon'(\varepsilon) $ resp.
$\varepsilon'' = \varepsilon''(\varepsilon) $,
and independent of the particular
choice of the configuration.
There is an exponential decay of correlations in the probability
$P^y$.
\endproclaim
We do not prove these general (but more or less straighforward
once (3.89),(3.90) was established) facts here.
\footnote{The above arguments give also only an outline
of the full proof that $P^y$ constructed by (3.86)
gives really a probability measure on $\ex$ with the properties stated above.
To interpret further this measure even as a Gibbs measure on $\ex$
i.e. as a Gibbs measure of the {\it original hamiltonian (2.2)\/} it
would be useful to have another general statement, whose proof is omitted
in this paper (however, we do not know
a suitable reference): A limit, for $\Lambda \to
\zv$,
of strictly diluted Gibbs measures in $\Lambda$
is a Gibbs measure on $\ex$.
%(at least for the case of the constant boundary conditions $y$).
Instead of such a general statement we
consider below only the more special limit over conoidal $\Lambda \to \zv$.}
\definition{Definition}
Say that a volume $\Lambda \subset \zv$ is {\it balanced\/} or conoidal
if for any admissible system in $\Lambda$,
$$ V(\gb) \subset \Lambda \Rightarrow \vv \subset \Lambda.$$
To have examples of balanced sets, take the sets
(``rectangular cup $\&$ cap glued together'';
compare (3.19) and (3.20))
$$\{t \in \zv: \dist(t,\partial \square)
\leq 1/2 \ \diam \square \}$$ where $\square$ is a cube in
$\zet^{\nu -1}$
and $ \partial \square $ denotes its boundary in
$\zet^{\nu -1} \subset \zv$.
\enddefinition
For balanced volumes, diluted and strictly diluted partition
functions (and corresponding Gibbs measures) are obviously the
{\it same\/} and the fact that the limit of {\it diluted\/}
Gibbs measures is (for $\Lambda \to \zv$)
a Gibbs measure
on $\ex$ is obvious.
\footnote{The general case of an arbirary volume $\Lambda$ is not
considered here.
It requires some more care concerning the ``stability'' of
considered finite volume measures with respect
to various boundary conditions.
Anyway, these and related questions must be
investigated in more detail also whenever a completeness
of the phase picture constructed by Main Theorem is studied.
We postpone this discussion to
a forthcoming paper.}
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) with some {\it mixed model\/} instead
of the abstract P.S. model.
This is the case of models with continuous spins ( studied in [DZ]) having
several ``potential wells'', for example. In these models,
the expansion around positive mass gaussians (approximating the regimes
of the potential wells )
of 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.
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\medskip
\bye