\input amstex
\documentstyle{amsppt}
\magnification1200
\NoRunningHeads
\NoBlackBoxes
\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\ce{\Bbb C}
\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\col{\operatorname{col}}
\def\sign{\operatorname{sign}}
\def\diam{\operatorname{diam}}
\def\dist{\operatorname{dist}}
\def\supp{\operatorname{supp}}
\def\Supp{\operatorname{Supp}}
\def\card{\operatorname{card}}
\def\conn{\operatorname{conn}}
\def\con{\operatorname{con}}
\def\Tr{\operatorname{Tr}}
\def\ext{\operatorname{ext}}
\def\loc{\operatorname{loc}}
\def\full{\operatorname{full}}
\def\inn{\operatorname{int}}
\def\innn{\operatorname{inn}}
\def\exxt{\operatorname{exxt}}
\def\out{\operatorname{out}}
\def\Ff{F_{\text{formal}}}
\def\Fp{F_{\text{phys}}}
\def\Fm{F_{\text{meta}}}
\def\ssubset{\subset\subset}
\topmatter
\title Contour Methods and Pirogov Sinai Theory
for Continuous Spin Lattice Models \endtitle
\author
Milo\v s Zahradn\'\i k \endauthor
\address MFF--UK
Charles University, Prague \endaddress
\email mzahrad@karlin.mff.cuni.cz
\endemail
\date January 20, submitted to AMS Dobrushin Memorial Volume
\enddate
\keywords Low temperature Gibbs states,
continuous and
discrete spin lattice models,
`multiple well' potentials, Gaussian partition functions,
perturbation theory around
a positive mass Gaussian field, contours,
Pirogov-Sinai theory,
cluster expansion of polymer models
\endkeywords
\subjclass 82A25
\endsubjclass
\abstract We developed an alternative
version of the \ps \ theory of low temperature
continuous spin models,
substantially simplified comparing to
the original approach \cite{DZ}. The new method
adopted here
is based on a transformation of a continuous spin model
into a suitable discrete spin model
(of a standard `abstract \ps' \ type). It uses a
`Gaussian transformation' (0)
for a multiple well potential $U$
and
allows us to use methods of
discrete spin theory \cite{PS}, \cite{Z} and \cite{ZR}.
\endabstract
\thanks The author thanks to Robert A. Minlos for useful
discussions on the subject of (classical and
quantum) continuous spin models, during which
the key idea (0) of this paper emerged.
\endthanks
\endtopmatter
\head { 1. Introduction}
\endhead
This is an outline of the new approach
to the problem originally solved in \cite{DZ} (see also
\cite{ZG} for its summary). We reduce the
generality of models treated in \cite{DZ} but
considerably simplify or modify the arguments of that paper.
The new approach is still based on
the same leading ideas
and main ingredients, namely: \roster
\item[0] elementary theory of
Gaussian fields with positive mass
Hamiltonians,
\item perturbation theory around positive mass
Gaussian fields,
\item \ps \ theory of models with discrete spin.
\endroster However, the exposition is now
organised quite differently
and is much simpler.
Theorems 0, 1, 2 (and 3) related to the items above
summarize our results.
The main novel feature is a representation
of a potential $U$ of the following type
$$\exp(-U(x_i)) = \sum_{q \in Q} \exp(- U^q(x_i))
+ w(x_i),
\;\hbox{where}\;|w(x_i)|:=\exp(-U^*(x_i)).\tag {0}$$
Here, $i = (i_1,\dots,i_{\nu}) \in \zv$,
$\nu \geq 2$ and $x_i \in \er$ is
a spin configuration in
the lattice point
$i$. Furthermore $Q \subset \er$
is supposed to be a
`uniformly discrete' (possibly
infinite) set; points $q\in Q$ label
translation invariant `locally
ground' states $\{x_i = q\}\ , \ q \in Q$ of $H$.
The terms $U^q(x_i)$ are
suitable {\bf{quadratic}}, positive definite,
approximations of $U(x_i)$ around
state $\{ x_i = q,\, i \in A \}$,
and $w(x_i)$ is a remainder, assumed
to be {\bf{small}}. See (17B) below.
The last condition is, essentially, the
only assumption on
the potential $U$. It is valid for many {\bf{low
temperature}} situations.
We will use
representation (0) for all $i \in \zv$.
The perturbative term $w(x_i)$ will be treated
below as an effect of a newly introduced
artificial {\bf{energy barrier}} \ $U^*(x_i) = - \log
|w(x_i)|$. Decomposition
(0) combines the ideas of high temperature
expansion and of Fortuin--Kasteleyn type
representation (used
for the study of Potts
models; see e.g. \cite{KS}).
One can visualise (0)
as an attempt to approximate a given `Gibbsian
hilly landscape' $\{\exp(-U(x_i))$, $x_i \in\er\}$
for a given potential $U$
by a {\bf{sum}} of Gaussian (i.e. logarithmically
quadratic) `hills'
$\exp(-U^q(x_i))$ where
$ U^q, q \in Q $ are suitably chosen
quadratic approximations of
$U$ around the points $q \in Q$ of its {\bf{local minima}}.
Note that if we wanted to have $w(x_i)$ positive,
it would impose rather strong restrictions
on $U$. In fact, as we will see
below, nonpositivity of $w(x_i)$ causes no
problems, and this general case
can be handled
practically without any change compared to the
case of positive $w(x_i)$.
\remark{Notes} 1.
I do not know whether decomposition (0)
of factor $\exp(-U(x_i))$ was used
before in rigorous statistical
mechanics of continuous spin models. It
greatly simplifies technicalities.
One could call it a `Gaussian transformation'
of the Gibbsian factor $\exp(-U(x_i))$ around
approximating family $\{\exp(-U^q(x_i))\}$.
One could use
an analogous expression as (0) also for
pair interactions, or for suitable `blocks'
of (both pair and single spin) interactions.
Throughout this paper,
we will not exploit such a more general approach. We
leave it for a further
study.
2. The special case of $w(x_i) \equiv 0$ offers substantial
simplifications. Here, the reduction to a discrete spin
model is particularly elegant. See the preprint
\cite{Ku}.
3. The phase transitions in
models with double well potentials were first studied
in \cite{GJS} (see also \cite{GJ}) and then in \cite{DS},
by methods of reflection positivity.
The paper \cite{I} first used the idea of
conversion to
discrete spin models (in a context very different to that
which is used here).
The paper \cite{DZ} (which arose independently of \cite{I})
developed a general machinery of \ps \ theory
for these models.
Now we propose a method which is
much simpler than these approaches (having in mind that a
developed theory for discrete spin models exists).
While the use of perturbative techniques
(e.g. estimates of semiinvariants of Wick polynomials)
around Gaussian fields is minimised
in our new approach, we recommend the
paper \cite{M} and the book
\cite{MM} for a detailed information
on these important techniques.
\head
2. The Hamiltonian and its Quadratic Approximation
on a Torus \endhead
The configuration space of the model under consideration
is $X= \er^{\zv}$. [In principle, one
could consider a configuration space $X = S^{\zv}$,
where $S\subset\er$ is a suitable
subset of $\er$, reasonably `dense' in those parts of
$\er$ where values of
$U$ are close to a local minimum.]
Similarly, for any finite torus $\Lambda$ (a quotient of $\zv$
with respect to a subgroup of a finite index)
we consider the configuration space $\er^{\Lambda}$.
We
prefer to work with these `periodical boundary conditions'
in contrast to the existing
tradition in the \ps \ theory. In fact,
the corresponding
formulas are simpler for the periodic than
for the case of a constant
boundary conditions $x_i\equiv q$ on $\Lambda^c$.
The problem of a transition from finite tori
to infinite volume limits deserves a further
study, in the general context of abstract \ps \
models.
We assume that the Hamiltonian of the model
is given, for $x_{\Lambda} = \{x_i, i \in \Lambda\}$ as
$$H(x_{\Lambda})=
H^{\text{pair}}(x_{\Lambda})+H^{\text{single}}(x_{\Lambda}) \
\ \text{with} \ \
H^{\text{single}}(x_{\Lambda})=\sum_{i \in \Lambda}
U(x_i). \tag 1 $$
\definition{Agreement} Writing the Gibbs factor
as $\exp(-H(x))$
instead of its usual form $\exp(-h/T\ H(x))$
we make an agreement
that the Boltzmann factor $h/T$
is included into the Hamiltonian. To avoid confusion
we should stress yet now that the
Peierls condition
proven below in Theorem 2 will be
derived from the condition (17B) (and (17C)) which will be
normally
fulfilled for {\bf{low enough}} temperatures
only!
Below we always work on some finite torus
$\Lambda$.
\enddefinition
The potential field $U$ is assumed to have a standard form of a
{\bf{multiple--well}}
$$ U(x_i)=\tilde U(x_i)\prod_{q\in Q}(x_i-q)^2 +
\hat U (x_i) \ , \
x_i \in \er. $$
Here $\tilde U (x_i)> 0$ varies smoothly
(and rather {\bf{slowly}}) between
the points of $q \in Q$ (which are
{\bf{local minima}} of function $x_i \in\er\mapsto
\tilde U(x_i)\prod_{q\in Q}(x_i-q)^2$),
and $\hat U (x_i)$ is small in
some $\delta_q$--neighborhood of $q$, where
$\delta_q$ is {\bf{not}} too small. Pictorially,
this equation can be written, in the $\delta_q$
neighborhood of
$q \in Q \subset \er$, as
$$ U(x_i) = U^q(x_i) +\text{small remainder ,
\ where} \ \ U^q(x_i) = d_q (y-q)^2+e_q
\tag 2$$
with some constants $d_q > 0$ and $e_q$.
(See more precisely (17B).)
An example of a multiple
well potential which is especially well suited
to a~ transformation (0) around the
temperature $T = 1$ is (see \cite{Ku})
$$U(x_i) =
-\log \bigl(\sum_{q \in Q} \exp(-d_q(x_i-q)^2 -e_q)\bigr)
.$$
The {\bf{pair interaction}} part of the Hamiltonian
is considered to be
$$ H^{\text{pair}}(x_{\Lambda}) = (Bx_{\Lambda},
x_{\Lambda}) =
\sum_{(i,j) \in \Lambda \times \Lambda} B_{i,j} x_i x_j,
\;\;x_{\Lambda}=(x_i, i \in \Lambda )\in \er^{\Lambda}. \tag 3$$
Here
$(\,\cdot\,,\,\cdot\,)$ denotes the scalar product in
$\ell_2(\Lambda)$, and
$(Bx_{\Lambda},x_{\Lambda})$
is a quadratic form in $\{x_i,\ i \in \Lambda\}$
given by a symmetric matrix $B$.
This quadratic form is moreover assumed to be
invariant with respect
to the space shifts of $x$ in $\Lambda$ and
also with respect to all shifts
$ x_{\Lambda} \to x_{\Lambda} + \text{const}$.
In
other words, we require that (3) equals to
$$ H^{\text{pair}}(x_{\Lambda}) =
\sum_{\{i,j\} \subset \Lambda}
b_{i-j} (x_i -x_j)^2 \tag 3' $$
for suitable
$b_i = b_{-i}$. Then
$B_{i,j}$ $=$ $ b_{i-j}$ $=$ $b_{j-i}$ $=$ $B_{j,i}$ for
$j \ne i $ and $B_{i,i} =\sum_j b_j$ for all $i$.
A finite-range condition $b_k$ $=$ $0$ for
$|k|>R$ will be also assumed.
In principle, we could also incorporate
(at the expense of simplicity) a
remainder $\hat H^{\text{pair}}(x)$
that is small
in a neighborhood of all
constant configurations $\{x_i = q\}$, $q \in Q$.
We add the diagonal part $D_q (x) =
\sum_{i \in \Lambda} d_q (x_i -q)^2$ (from (2))
to $(Bx_{\Lambda},x_{\Lambda})$.
In sections II and III, we moreover put $e_q \equiv 0$
for the simplicity of notations.
Write
$$H^q(x_{\Lambda}) = (Bx_{\Lambda},x_{\Lambda})+
D_q(x_{\Lambda})=\sum_{\{i,j \}\subset \Lambda}
b_{i-j}(x_i -x_j)^2+\sum_{i\in\Lambda}d_q(x_i -q)^2
. \tag 4$$ Equivalently, $H^q(x_{\Lambda})$
can be also defined as
$(A\tilde x_{\Lambda},\tilde x_{\Lambda})$
$=$ $\sum_{i,j\in \Lambda}A_{i,j}\tilde x_i \tilde x_j$ \
where $\tilde x_i=x_i -q$ and
the symmetric matrix $A$
has entries $A_{i,j}$ $=$ $\alpha_{i-j}$, with
$$\alpha_{k}= -b_{k},\;k\neq 0,\;\;\alpha_0 =
\alpha^q_0
=d_q+\sum_{\tilde k:\;\tilde k\neq 0}
b_{\tilde k}.$$
We can interpret
the operator $\{\tilde x_{\Lambda} \mapsto A\tilde
x_{\Lambda}\}$ as
a {\bf{convolution}} $\{x_{\Lambda}
\mapsto \alpha*x_{\Lambda}\}$, where $(\alpha*x_{\Lambda})_i
=\sum_{j \in \Lambda} \alpha_j x_{i-j}$
has a symmetric convolution kernel
$$
\alpha = \alpha^q = \{\alpha^q_k
= - b_k, \ \alpha^q_0 = d_q + \sum_{j \ne 0}
b_j \}. $$
Notice that with $d_q > 0$ the form $(A \tilde
x_{\Lambda}, \tilde x_{\Lambda})$ is {\bf{positive definite}}
which means that the Fourier transform
$\hat \alpha^q(\lambda)$
$=$ $\sum_j \exp(2\pi i (\lambda, j))\alpha^q_j$
is positive for each
$\lambda \in [0,1]^{\nu}$.
When necessary, write
$A^q$ instead of $A$ thus stressing its dependence
on $\alpha_0^q$.
\definition{Assumption} In the following, we will assume that
$$ \sum_{j \ne 0}|\alpha_j^q| \leq \kappa^q \, \alpha_0^q \ \
\ \ \ \text{where} \ \ \kappa^q \ll 1 .\tag {5}$$
\enddefinition
\proclaim{Proposition}
Any positive definite, finite range convolution
kernel
$\alpha$ defined above
satisfies the property (5) when expressed in suitable
new
coordinates $\xi_{\Lambda}$. The required
change of coordinates
is given by a
convolution with a symmetric kernel
$$ \xi_{\Lambda} = K * \tilde x_{\Lambda} \ \ ; \ \
K = \{K_i ; \ |i| \leq R'\}. \tag 6 $$
\endproclaim
\demo{Proof} It suffices to take
$K$ suitably close to the (infinite range)
kernel $K^{\infty} =\sqrt\alpha$
diagonalizing the quadratic form $(A \tilde
x_{\Lambda}, \tilde x_{\Lambda}) = (\alpha * \tilde x_{\Lambda},
\tilde x_{\Lambda}) =
(K^{\infty} * K^{\infty} * \tilde
x_{\Lambda},
\tilde x_{\Lambda })= (K^{\infty} * \tilde
x_{\Lambda},
K^{\infty} * \tilde x_{\Lambda }$) .
The terms of the kernels $K^{\infty}$ and $(K^{\infty})^{-1}$
decay
more quickly than the inverse of any polynomial of $i$.
Really,
the Fourier transform of $K^{\infty}$
is an infinitely differentiable positive function. (It is the
square root of the above trigonometric polynomial
$\hat \alpha^q(\lambda)$.)
We now cut
suitably $K^{\infty}$
by taking
$K_i =K^{\infty}_i$ for $ |i| \leq R'$ and
$K_i = 0 $ otherwise.
We put $\hat K = K^{\infty } - K$.
We have $$ (A \tilde x_{\Lambda}, \tilde x_{\Lambda})
=
(\alpha * \tilde x_{\Lambda}, \tilde x_{\Lambda}) =
( K^{-1} * \alpha * K^{-1} \xi_{\Lambda} , \xi_{\Lambda} )
= ( \xi_{\Lambda} , \xi_{\Lambda} )
- (\Delta * \xi_{\Lambda}, \xi_{\Lambda})
$$ where
$ \Delta= K^{-1}* \hat K + \hat K* K^{-1}
- K^{-1}*\hat K * \hat K * K^{-1}$.
Then we can write $K^{-1}$ as
$K^{-1} = (K^{\infty} -\hat K)^{-1} =
(K^{\infty})^{-1} *
(J +\sum_{n=1}^{\infty} L^n )$ where
$ L = \hat K *(K^{\infty})^{-1} $.
Notice that
$\hat K_i =0$ for $|i| > R'$ and so
$L$ decays like $|L_i| \leq \varepsilon_{L,d}\ |i|^{-d}$
with a small constant $ \varepsilon_{L,d}$ and
$d > \nu$.
It is then easy to show also that $ |L^2_i| \leq
\varepsilon_{L^2,d} \ |i|^{-d}$ with
$\varepsilon_{L^2,d} \ll \varepsilon_{L,d} $ etc. and
analogously that
$|(\sum_nL^n)_i| \leq \varepsilon\ |i|^{-d}$ with a small
$\varepsilon = \varepsilon_{d,R'}$.
Thus one establishes (5) for
$J -\Delta$ ($J$ identity operator),
for enough large $R'$.
\enddemo
The fact that $K$ in (6) has a {\bf{finite}}
range is important.
Namely, the mapping
$\tilde x_{\Lambda}
\mapsto
\xi_{\Lambda} = K * \tilde x_{\Lambda} $
increases the range of interactions of the
model, from the value $R$ to the value $R +R'$.
In fact, the value $R+ R'$ must then be used -- instead
of $R$ -- in the definition
of precontours!
For reasons of simplicity, we study below
{\bf{only}} the case when (5) is directly valid;
like in our canonical {\bf{example}}
$$ H^q(x_{\Lambda}) = (A \tilde x_{\Lambda},
\tilde x_{\Lambda}) = \sum_{\{i,j\} \subset \Lambda:
|i-j| \leq 1}
b_{j-i}(\tilde x_i - \tilde x_j)^2
+ d_q \sum_{i\in \Lambda} (\tilde x_i)^2 \ ,
\ \tilde x_i = x_i -q
$$
giving the value
$ \kappa^q = { \sum_j |b_j| }/{ (\sum_j b_j +d_q)}
\ll 1$ for a suitably big $d_q$.
\head 3. Auxiliary results for Positive Definite
Quadratic Hamiltonians
\endhead
Because of the special form of our quadratic Hamiltonians
(defined as Toeplitz quadratic forms $(Ax,x)$
with a
finite range convolution kernel; we put $q = 0 $ $e_q =0$
everywhere in this chapter and
we do not indicate
the dependence of $A$ on $q$ here)
$$H(x_{\Lambda}) = (Ax_{\Lambda},x_{\Lambda}) \ \
\text{where} \ \ Ax_{\Lambda} = x_{\Lambda}*\alpha \ ;
\ \ \alpha = \{\alpha_t ,\ t \in \Lambda \}\ ,
\ \alpha_t = \alpha_{-t} $$
the results stated below will be
simpler than that of \cite{DZ}.
The Hamiltonian $H(x_{\Lambda})$
will denote a purely quadratic function everywhere
in this chapter.
\definition{Expression of the partition function on torus}
\enddefinition
Denote by $\mu_{\Lambda}$ the Gaussian measure
on $R^{\Lambda}$ ($\Lambda$ is a torus)
having the density
$$ \frac{d \mu_{\Lambda} (x_{\Lambda})}{
d x_{\Lambda}} =
\frac{\exp(-H(x_{\Lambda}))}{Z(\Lambda)} \ \ ;
\ \ \ Z(\Lambda) =
\int_{\er^{\Lambda}} \exp(-H(x_{\Lambda})) \ d x_{\Lambda}
\tag{7} $$
where the Gaussian partition
function is expressed as
$ Z(\Lambda)
= \pi^{\frac{|\Lambda|}{2}} (\det A)^{-1/2}$
i.e.
$$ 2 \log Z(\Lambda) - \ \log \pi \ |\Lambda| = -
\log \det A =
- \Tr(\log A) =
- \Tr \log(I - W) - \log \alpha_0 |\Lambda| \tag 8$$
where $I$ is the identity matrix and
$ W = I - A / \alpha_0 $ has zero diagonal.
We are using here the
formula $ \det \exp \tilde A
= \exp \Tr \tilde A$, for $ A = \exp \tilde A$.
This can be written, with the help of the
transition weights $w_{ij} = w_{i-j}$ given,
for any $i \in \zv, i \ne 0$, as
$$ w_i = {\alpha_i}/ {\alpha_0} \ \ \ \ \ \ \ \ \
(\text{where} \ \ \ \sum_{i \ne 0} |w_i| < \kappa < 1 \ )
$$
in the following way, just by taking
the Taylor expansion of the logarithm
of $I -W$
$$ -\Tr \ \log(I - W) = \sum_{n\geq 2} 1/ n \Tr W^n =
\sum_{P} w_P \ \ \ \text{where} \ \
w_P = \prod_{\{i,j\} \in P} w_{j-i} \tag {9}$$
and where the last sum is over all
{\bf{ closed }} oriented paths
\footnote{Notice that the term $1/n$ disappeared
in $\sum_P w_P$
(because each $P$ is taken $|P|$ times
there).}
in
$\Lambda$ with `steps' $(i,j)$ such that $i \ne j$.
Define then the {\bf{free energy}} of this Gaussian model
(the sum being taken over all closed paths $P$
containing a selected point, say $0$)
$$ -h = 1/2 \ \bigl(\log \pi
-\log \alpha_0 + \sum_{P \owns 0} \frac{w_P}{|P|}\
\bigr) .\tag {10}$$
\definition{Computation of the
conditioned minima for quadratic Hamiltonians}
\enddefinition
In this section,
we develop some further preparatory material, to be summarised
later in Theorem 0.
Let $M\subset \Lambda$.
The forthcoming formulas are valid
even for an infinite $\Lambda$,
however
$M^c \equiv \Lambda \setminus M$ has to be finite.
Denote by $\bar x_M$ the configuration
{\bf{minimising}} $H(x_{\Lambda})$
under a fixed $x_{M^c} = \bar x_{M^c}$. An
elementary computation (consider the derivative of the
expression
$f(t) = H(t \bar x_i \cup \bar x_{M\setminus i} \cup
\bar x_{M^c})$
and put $f'(1) = 0$ !) shows
that the relation $ \bar x_i = (W \bar x )_i$
holds for each
$i \in M$. Then, by {\bf{iterating}} this expression
(until reaching values $\bar x_i$ on the set $M^c$),
we can express the values $\bar x_i, i \in M$
and also the value of the quadratic
form $ (A\bar x_{\Lambda},\bar x_{\Lambda}) =
\alpha_0 ((I -W)\bar x_{\Lambda}, \bar x_{\Lambda})$
by the following formulas
(the weights $w_P$ are given by (9))
$$ \bar x_i =
\sum_P w_P\ \bar x_{e(P)} \ \ \text{and} \ \
(A\bar x_{\Lambda}, \bar x_{\Lambda}) = \alpha_0 \ \bigl(
\sum_{i \in M^c} x_i^2 - \sum_P w_P \
\bar x_{s(P)}\bar x_{e(P)} \bigr). \tag 11 $$
Here, the first sum is over all
(nonclosed) oriented paths $P$ starting
in $i \in M$ and reaching the set $M^c$ only at its
{\bf{end point}}
$e(P)$.
The second sum
in the expression of $(A \bar x, \bar x)$
is over all oriented paths $P$
intersecting $M^c$ {\bf{ exactly twice}}:
at the starting point
$s(P)$ and at the end point $e(P)$.
The rest of the
path $P$ (possibly empty) is spent in $M$.
To derive the
{\bf{second formula}} in (11) from the first one,
substitute the above expression (
11)
of $\bar x_i$ into $(A \bar x_{\Lambda}, \bar x_{\Lambda})$
= $\alpha_0 ((I -W)\bar x, \bar x)$:
1)
Consider first the $-(W \bar x, \bar x)$ term i.e.
all the possible decompositions
of the non oriented path $P$ (from $M^c $ back to $M^c$)
into a triple
$P = P_1 \cup b \cup P_2, \ b = \{s(P_1),s(P_2)\}$
with oriented paths $P_1, P_2$ going from $s(P_i) \in M$
to $M ^c$ or being empty. Each
such triple contributes twice the minus
sign.
2) One then has to add, to
$-(W \bar x, \bar x)$, also the $(\bar x,\bar x)$
term which can be expressed
as the contribution, with the coefficient $2$
($1$ in the case when $P_1 = P_2$),
of all possible
pairs $P = P_1 \cup P_2$ of oriented
paths $P_1,P_2$ going from a point
$s(P_1) = s(P_2) \in M$ to $M^c$.
Taking in account the signs of all these terms,
the net
contribution of a path $P, |P| = n $, is equal
to $ ( -2n + 2(n-1)) w_P = -2 w_P$
(and similarly for the special case when $ P_1 = P_2$
for some $P_1 = P_2$). So it is equal to
$- w_P$ if we consider the {\bf{two}}
possible
orientations of such a path $P$.
\definition{Conditioned Gaussian fields,
the correlation decay. Summary}
\enddefinition
The forthcoming Theorem 0 summaries everything what we
will need to know on the partition functions of Gaussian
Gibbsian fields defined by above quadratic
forms $(Ax_{\Lambda},x_{\Lambda})$
where $A = \alpha_0 ( I -W)$),
and on the correlation decay of the corresponding
conditioned Gaussian fields.
We remark in advance
that the formulas (13), (13') and (14)
below are valid universally
(with weights $w_P$ given by (9))
but they will be really useful later
only under the condition (5).
Given a subset $M$ of a torus $ \Lambda$
consider the `conditioned' Hamiltonian
$$ H(x_M|x_{M^c}) =
H(x_{\Lambda}) -H(x_{ M^c}\cup 0_M) =
\sum_{i,j \in \Lambda: \{i,j\} \cap M \ne \emptyset} A_{i,j}
\, x_i x_j \tag 12 $$ defined for any
$ x_{\Lambda} = x_M \cup x_{M^c} \in \er^{\Lambda}$.
For $q\ne 0 $, one has to write $q_M$ instead
of $0_M$ and $x_i- q$ instead of $x_i$
in the definition of $H^q(x_M|x_{M^c})$.
Introduce also the {\bf{conditional probabilities}}
$\mu_{\bar x_{M^c}}(x_{M})$, by (7).
We note that an alternative to (12)
$$\tilde H^q(x_M|x_{M^c}) = \tilde H^q(x_{\Lambda}) -
\tilde H^q(x_{M^c})
=
\sum_{\{i,j \}\nsubseteq M^c}
b_{i-j}(\tilde x_i -
\tilde x_j)^2+\sum_{i\in M}d_q(\tilde x_i )^2 $$
where $\tilde x_i = x_i -q$ and $ \tilde H^q(x_M) =
\sum_{\{i,j \}\subset M}
b_{i-j}(\tilde x_i -
\tilde x_j)^2+\sum_{i\in M}d_q(\tilde x_i)^2$
will be more natural in
section 8 but not here
where we are working
with the quadratic form $(Ax_{\Lambda},x_{\Lambda})$.
\footnote{Notice that $H^q(x_M| x_{M^c}) $
is nonzero, in contrary to
$\tilde H^{q}(x_M|
x_{M^c})$, for a constant $x_{\Lambda} \ne q_{\Lambda}$.
A confusion could arise here as
the concepts of `pair' and `single spin'
interactions change its meaning
when expanding
$\sum b_{i-j}(x_i -x_j)^2 = (\sum_j b_j) \sum (\tilde x_i)^2
-2\sum b_{i-j} \tilde x_i \tilde x_j$.}
However, we have
$\tilde H^q(x_{\Lambda} )\equiv H^q(x_{\Lambda})$.
We have also the relation
$$H^q(x_M \cup q_{M^c}) =
H^q(x_M|\, q_{M^c}) = \tilde H^q(x_M) +
\tilde H^q(q_{M^c}|\,x_M)
= (A \tilde x_M,\tilde x_M). $$
\proclaim{Theorem 0} Assume (5). Write again
$H^q \equiv H$ and put $q= 0, e_q =0$.
Denote by $\bar x_M$ the configuration
minimising $H(x_{\Lambda})$ under
$x_{M^c}
= \bar x_{M^c}$.
Then
$ H(x_M \cup \bar x_{M^c}) - H(\bar x_{\Lambda})
= H(x_M|\bar x_{M^c}) - H(\bar x_M|\bar x_{M^c})
= H((x-\bar x)_M \cup 0_{M^c})
$ and
$$Z(M | \bar x_{M^c}) = \int_{ \er^M}
\exp (-H(x_M|\bar x_{M^c})) \, d x_M =
\exp(-H(\bar x_M|\bar x_{M^c})) Z(M| 0_{M^c} ) . \tag 13$$
Moreover we have the following expression of the
right hand side of (13) in terms of random walks,
with the same notations as in (9) and (12), see also (11)
$$ H(\bar x_M|\bar x_{M^c}) =
- \alpha_0 \ \bigl( \sum_{P:
P \cap M \ne \emptyset} w_P \
\bar x_{s(P)}\bar x_{e(P)}
\bigr)\tag 13' $$
where the sum is over all oriented paths
$P$ from $M^c$ to $M^c$ crossing (at least in one point)
the set $M$. The partition function
$Z(M|\, 0_{M^c})$ is expressed as
the sum over all oriented paths $ P$
{\bf{not}} intersecting $M^c$, resp. intersecting
{\bf{both}} $M$ and $M^c$
$$ \log Z(M| \, 0_{M^c} \ ) = 1/2 \ \bigl( (\log \pi -
\alpha_0)\ |M|
+ \sum_P w_P \bigr) = -h|M| - \sum_{\tilde P}
\tilde w_{\tilde P}
\tag {14} $$
where the free energy $h$ is from (10) and
$\tilde w_P$ are related to $w_P$ (see (9)) as
$$ \tilde w_P = \frac{| P \cap M|}{2 | P|} w_P \ ; $$
this notation will be used also below.
The symbol $|P \cap M |$ denotes the number of visits
of $P$ in $M$.
The first sum in (14) is over all closed oriented paths
$P$, below called {\bf{ loops}}
(in comparison to
the {\bf{semiloops}} $P$ of (13')) in $M$.
The second sum is
over loops $\tilde P$ intersecting both $M$ and
$M^c$.
The values $\bar x_i, \, i \in M $ are expressed by
(11) and estimated, for the case of
interaction range \footnote{
We will not optimise this bound. After all, (11)
is more detailed.} $R$
$$ |\bar x_i| \leq (1- \kappa)^{-1}
\kappa^{\frac{\dist(i,M^c)}{R}} \, \parallel
\tilde x_{M^c} \parallel_{\infty}
.\tag 15$$
\endproclaim
\demo{Proof} This is just a recapitulation of the arguments
used when deriving (8)--(11).
The relation $H(x_M \cup \bar x_{M^c}) =
H(\bar x_M \cup \bar x_{M^c})
+ H(x_M -\bar x_M |\, 0_{M^c}) $ follows
easily
if we notice
that $H(t (x_M - \bar x_M ) + \bar x_{\Lambda})$
attains its minimum (as a function of $t \in \er$)
at $t = 0$.
The sum over paths in the formula (14)
emerges like in (9):
$$2 \log Z(M|\ 0 \ ) -(\log \pi-\log \alpha_0) |\Lambda| =
- \log \det (I - W_M) =
-\Tr \log (I -W_M) , $$
where $ W_M = I -A_M/ \alpha_0$ . This is a notation
analogous
to that of (9) and the symbol $A_M$ denotes the quadratic
form $(Ax,x)$ restricted to the configuration space $R^M$.
Finally we have the expression, with the same weights $w_P$
as in (9)
$$ - \Tr \log(I - W_M) = \sum_n 1/ n \ \Tr W_M^n =
\sum_{P} w_P \tag {16}$$ where the sum is over all closed
oriented
paths {\bf{in the
volume}}
$M$. The expression of $H(\bar x_M|\bar x_{M^c})$
used in (13') then follows from (11) and (12).
Namely, by (12) we have the
relation
$ H(\bar x_M|\bar x_{M^c}) =
(A\bar x_{\Lambda}, \bar x_{\Lambda}) - (A \bar x_{M^c},
\bar x_{M^c})$.
Then we can express
$(A\bar x_{\Lambda}, \bar x_{\Lambda})$ by (11)
and subtract the term $(A \bar x_{M^c},
\bar x_{M^c})$ from it. Only contributions of
nontrivial paths `crossing' $M$ will remain.
Finally, to get
(15) one notices that the summation over
all paths $P$ from $i$ to $M^c$ gives
$$ \sum_{P} |w_P| \ \leq \ \sum_{k: \ R k > \dist(i, M^c)}
\ ( \sum_{j: |j| \leq R} |w_j| )^k < \
(1-\kappa)^{-1} \kappa^{\, \frac{\dist( i, M^c)}{R}}
. $$
\enddemo
\head { 4. Conditions on the Potential $U$. }
\endhead Now we are again studying a
general nonquadratic
Hamiltonian (1). We will discuss in more detail
the decomposition (0). Recall that we assume
(for simplicity)
that
the {\bf{pair}} interactions in (1) are
purely {\bf{quadratic}} ones.
\footnote {For nonquadratic
pair interactions,
a similar expansion to (0) can be applied in principle.
Usually,
one would have only one
quadratic term there but the method applies
also to
models with more complicated interactions
$\Phi_{\{i,j\}} (x_i,x_j) =
\tilde U(x_i -x_j)$ where $\tilde U$ has
{\bf{several}}
local minima. }
Recall that we take the decomposition (0) with suitable
$U^q(x_i) = e_q + d_q (x_i-q)^2$ (see (2)).
The approximations
$U^q$ are now with $e_q \ne 0$ in general.
In contrast to the `temperature independent'
considerations of the preceding sections
the fact that
we have a {\bf{sufficiently low}} temperature
will be crucial now, and this will be expressed
in terms of the `potential'
$U^*(x_i) = - \log |w(x_i)|$
assumed to be enough large:
\definition{Assumptions on $U^*(x) $ and the approximating
Hamiltonians $H^q$} \enddefinition
Recall that around each $q \in Q$ we have the
approximate relation
$ U(x_i) \doteq U^q(x_i)$.
More precisely see (17B) below (we
will in fact formulate such a requirement
in conjunction with the condition on the
`sufficient height of the hills' of $U$,
between all the pairs of nearest neighbors $q < q'$ in
$ Q\cup \{\infty\}
\cup \{-\infty\} $). Our formulation will be given
directly in
terms of $U^*$.
(After all, if we are outside of
the `wells' of $U$ then either both $U$ and $U^*$
are `safely big'
or we have
$U^* \doteq U$. Then we may write $U$ in (17B).)
Denote by
$\delta_q \equiv 1/2 \,
\dist(q , Q\setminus \{q\}) $ and assume that the
quantity
$$ \delta =\min_{q \in Q} \delta_q = \frac{1}{2}
\ \min_{q,q' \in Q :\ q \ne q'} |q' -q| \tag 17A $$
is not too small (see (17C) below). We need a precise
condition on
`sufficient energy barriers':
Introduce an
auxiliary function
$ \rho(x_i) = \min \ \{ \rho_q (x_i) , \rho_{ q'}(x_i) \} $
in any interval $ (q , q' ) $ where
$q < q'$ are nearest neighbors from
$ Q \cup \{\infty\}
\cup \{-\infty\}$ and
$$
\rho_q (x_i) = ( \log_2 (1+ |x_i-q|/ \delta_q) +1)^{\nu}
\ \ \ \ (\rho_{\pm \infty} \equiv \infty)
$$
Assume that for each $q < r < r' < q'$ such
that $\rho(r) = \rho(r') \geq d$
(we take $r', r = \pm \infty $ if $q',q = \pm\infty$)
we have a
bound, with a {\bf{ small}}
$\varepsilon$
$$ \int_r^{r'} \exp(-U^*(x_i)) \, d x_i\
< \ \varepsilon^{d}\ .
\tag 17B$$
The conditions (17A) and (17B)
summarise our requirements on $U$.
\remark{Note} The condition
(17B) gives also a precise meaning
to the statement $ U(x_i) \doteq U^q(x_i)$ used
in the vicinity of $q$.
All the `important'
(taking in account also the widths of
the corresponding `wells') local minima
of $U$
should be represented
by some
$q \in Q$. Usual examples of $U$ with
`well separated' local minima $q \in Q$ (even e.g.
$U(x_i) = \sin \pi x_i$, $Q = \zet$) satisfy (17B)
for enough low temperatures,
and it suffices to check (17B)
for $r= q$ and $r' = q'$.
In fact, (17B) is an energy/entropy
contest
condition and to have an idea how
it could be violated
imagine that
the `hill' of $U^* (\doteq U)$ between $q $ and
$q'$ looks
like a `low and extremely long plateau'.
\endremark
Conditions (17 A,B) will enable to prove
the validity of the {\bf{Peierls condition}},
if complemented by another
assumption of positive definiteness, `
on $(1,1,\dots,1)^{\perp}$',
of the pair Hamiltonian $H^{\text{pair}}(x_{\Lambda})
\equiv \sum_{\{i,j\} \in \Lambda}
b_{i-j} (x_i -x_j)^2 $: Assume that
$$ H^{\text{pair}}(x_{\Lambda}) >
\alpha \sum_{i \in \Lambda}
\sum_{j: |j-i| =1} ( x_i -x_j)^2
\tag 17C $$ for some $\alpha$ such that
$\alpha \delta^2$ (see (17A))
is enough {\bf{large}}. (This is
an assumption on low
temperature, too. Namely $\alpha$ is proportional to
the inverse temperature $\beta$.)
Let us go back to the Hamiltonian $H^q$.
From (5)
we get the following bound for $H^q(x_{\Lambda})$.
Below we assume that also all $\alpha^q_0$
are enough large: \footnote{Notice
that $\alpha_0^q -d_q \asymp \alpha $
with $d_q > 0$ (see (2)); $\delta$ is temperature
independent.}
$$ \alpha_0^q (1+ \kappa^q) \sum_{i \in
\Lambda} |x_i -q|^2 \ \geq \
H^q(x_{\Lambda}) - e_q |\Lambda| \ \geq \
\alpha_0^q (1- \kappa^q)
\sum_{i \in \Lambda} \ |x_i -q|^2 . \tag 17D $$
For any $\Lambda \supset \tilde M \supset M$
and any configuration $x_M$ consider the
quantity $$ H^q_e(\tilde M| x_{M})
= \min_{ x_{\Lambda}} (H^q(\tilde x_{\Lambda})
-e_q|\Lambda|)$$
where the minimum is
over all extensions
$ x_{\Lambda}$
of $x_{M}$ such that
$ x_i = q $ for each $ i \in (\tilde M)^c$.
Then we have a bound\footnote{See
(13') for more detailed information on
$ H^q_e(\tilde M|x_M)$. Hence
estimates of $C^q$ and $\tilde C^q$ .},
more useful below than (17D)
$$ C^q\alpha_0^q
\sum_{i \in M} \ |x_i -q|^2
\geq \ H^q_e(\tilde M| x_{M})
\ \geq \tilde C^q \alpha_0^q
\sum_{i \in M}
\ |x_i -q|^2 \tag 17E$$
with suitable constants $C^q = C^q(\kappa^q) \doteq 1$
and $\tilde C^q = \tilde C^q(\kappa^q) \doteq 1$.
\head{5. Expression of
$Z(\Lambda)$ using (thick)
precontours }
\endhead
Write the partition function of the model with Hamiltonian (1)
$$
Z(\Lambda) = \int_{\er^{\Lambda}}
\exp (-H(x_{\Lambda})) \ d x_{\Lambda} \tag 18
$$ by inserting the expression
(0) into each Gibbs factor
$$ \exp(-H(x_{\Lambda})) = \prod_{ \{i,j\}} \exp(- b_{i-j}
(x_i -x_j)^2 ) \prod_{i \in \Lambda} \exp(-U(x_i)). $$
Denote $Q^* = Q \cup\{*\}$, i.e. enrich
the reference set $Q$ by another
value $*$ in the expression (0). In other words,
write $ \exp(-U(x)) = \sum_{q \in Q^*}
\exp(- U^q(x))$. Then expand all the parentheses.
We get the following
formula.
\proclaim{Proposition 1} The partition function $Z(\Lambda)$
can be expressed as
$$ Z(\Lambda) = \sum_{y =y_{\Lambda} \in (Q^*)^{\Lambda}}
\int_{\er^{\Lambda}}\exp(-H_y(x_{\Lambda}) )
\prod_{i: y_i = *} w(x_i) \ d x_{\Lambda}, \tag 19$$
where the (translation noninvariant!) Hamiltonian
$H_y(x_{\Lambda})$
is given by the formula
$$
H_y(x_{\Lambda}) = \sum_{\{i,j\} \subset \Lambda }
b_{i-j}(x_i -x_j)^2
+ \sum_{i \in \Lambda: y_i \ne
*} U^{y_i}(x_i)
$$
and where the summation and integration is over all
pairs $(x_{\Lambda},y_{\Lambda})$ composed
of a {\bf{configuration }}
$x_{\Lambda} \in \er^{\Lambda}$ and a {\bf{`coloring'}}
($Q^*$--valued
configuration)
$y_{\Lambda} \in (Q^*)^{\Lambda}$.
\endproclaim
\demo{Proof} This is just
the result of expanding the parentheses
that appear when (0) is inserted into (18).
Of course, the leading idea here is that the most important
contribution to $Z(\Lambda)$
comes only from
those pairs $(x_{\Lambda},y_{\Lambda})$ where
the coloring $y_{\Lambda}$ `mimics, as best as possible'
the configuration $x_{\Lambda}$ -- but we
emphasise that the integral
and the sum here is
really
over {\bf{all}} the pairs
$(x_{\Lambda},y_{\Lambda}) \in \er^{\Lambda} \times
(Q^*)^{\Lambda}$.
\enddemo
In the following, we impose another simplifying
{\bf{assumption}}
namely that $R = 1$. It is straightforward to generalise
the considerations
below to the general case of interactions having
a range $R$ (resp. $R +R'$ if
a transformation (6) is used).
\footnote{One has to start with writing
$|j-i| \leq R$ in the
definition below.
However, the requirements on temperature
are much stronger for large $R$.
The case of fast decaying
infinite range interactions
could be also dealt with, like in \cite{YP}.
The proper choice of
a value $R$ (interactions of a range bigger than $R$ are then
treated as small perturbations) deserves
a detailed discussion.}
\definition{Precontours} \enddefinition
Say that $i \in \Lambda$ is a $q$--correct point
of $y_{\Lambda}$,
$q \in Q$,
if $y_j = q$ holds
for all $j$ such that
$|j-i|\leq 1$. Say that
$i \in \Lambda$ is {\bf{incorrect}} if
it is
$q$--correct for no $q \in Q$.
Thus, incorrectness of $i$ means
{\bf{either}} that $y_i$ has the value $*$ {\bf{or}} that
the value $x_i= q$ is not shared by all
nearest neighbours of $i$.
It is
a property of the `coloring' $y_{\Lambda}$.
A restriction of $x_{\Lambda}$
to a connected component $G$ of the collection of all
incorrect points of $y_{\Lambda} \in (Q^*)^{\Lambda}$
will be called a {\bf{precontour}} of the
pair $(x_{\Lambda},y_{\Lambda})$, denoted by a symbol $\gamma$.
The set $G$ will be called the {\bf{support}}
of $\gamma$, denoted by $G = \supp \gamma$.
The configuration $y_G$,
denoted more precisely also as $y^{\gamma}_{G}$
is the {\bf{colour}}
of $\gamma$ (it can be extended uniquely to $\Lambda$
such that all points of $G^c$ are correct)
while $x_i, i \in G $ are the
{\bf{values}} of $\gamma$.
Define the {\bf{Gibbs weight}} of a precontour
$\gamma = (x_G, y_G)$
$$ w_{\gamma} = \prod_{i \in \supp \gamma: y_i \ne *}
\exp(-U^{y_i}(x_i)) \prod_{i \in \supp \gamma: y_i = *}
w(x_i)
\ \ \exp(-H^{\text{pair}}(x_G \cup y_{G^c}))
\tag {20}$$
where \
$H^{\text{pair}}(x_M \cup x_{M^c})
=\sum_{\{i,j\} \subset \Lambda}
b_{i-j} (x_i -x_j)^2$ is from (3').
Thus,
$w_{\gamma}$ depends both on $x_G$ and $y_{G\cup
\partial G^c}$.
(While the
support of a precontour is wholly determined by
the colour $y$, the value $w_{\gamma}$, possibly negative,
depends
also on the values
$x_{\supp \gamma} \in \er^{\supp \gamma}$ of $\gamma$.)
Notice that the colour of $\partial \supp \gamma=
\{i \in \supp
\gamma : \dist(i, \supp \gamma^c) = 1\}$ is locally constant,
taking some value $q \in Q$.
Say that $\{\gamma_n\}$ is an {\bf{allowed collection}}
of precontours if it exhausts all precontours
of some pair $(x_{\Lambda},y_{\Lambda})$.
Clearly, the property of being allowed
is determined only by the properties of
the coloring $y_{\Lambda}$
on the boundary of the union $\cup_n \supp \gamma_n$.
Outside that
set, the corresponding `correct' values of $q \in Q$ are
determined uniquely. For any $q \in Q$ we denote
by $\Lambda^q =
\Lambda^q (\{\gamma_n\}) = \{i \in
(\cup_n \supp \gamma_n)^c: y_i = q\}$ the
collection of {\bf{all
$q$--correct points}} of the allowed collection
$\{\gamma_n\}$ ($\equiv$ the collection of all
$q$--correct points of the
colour $y^{\{\gamma_n\}}_{\Lambda}$ induced by
$\{\gamma_n\}$ on $\Lambda$).
Denote
the values of $\gamma_{n}$
in $i \in
\partial (\Lambda^q)^c$ below as $x_i$.
We then have the following
\proclaim{Proposition 2}
The partition function
$Z_{\Lambda}$ is expressed, using (20), as
$$ Z(\Lambda) =
\sum_{G, y_{\Lambda}} \int_{\er^ G }
\prod_n w(\gamma_n)
\prod_{q \in Q} Z^q(\Lambda^q| x_{\partial
(\Lambda^q)^c}) \ d x_{G} \ \ \ \ \ \text{where}
\tag 21$$
$$ Z^q(M| x_{\partial
(M^c}) = \int_{\er^{M}}
\exp (-H^q(x_{M}| x_{\partial M^c}) \, d x_M.
\ \
(\text{We have} \
\gamma_n = (x_{\supp \gamma_n}, y_{\supp
\gamma_n}.) $$
The sum is over all possible
$G = \cup_n \supp \gamma_n$ and
$y_{\Lambda} = y^{\{\gamma_n\}}_{\Lambda}$.
The partition function
$Z^q(\Lambda^q|x_{\partial (\Lambda^q)^c}) $
is
a Gaussian one.
The integration $d x_G$ is over all $x_i \in \er,\ i \in G$.
\endproclaim
\demo{Proof} See the relation (19) and the notation (20).
We will denote the relative Hamiltonian
$H^q(x_{\Lambda^q}|x_{(\Lambda^q)^c})$
below also as $H^q(x_{\Lambda^q}|x_{\partial
(\Lambda^q)^c})$. Notice that we have a disjoint
decomposition
$\Lambda = G \cup \cup_q \Lambda^q$.
Using the correctness of $y$ outside of
$G$ we get the
following relation
\footnote{While using (12),
do not attemt to introduce any $H^{\text{pair}}(x_M|x_{M^c})$.
This would cause confusion.}
which obviously proves (21) from (19)
$$ \tilde H^Q_y(x_{\Lambda}) =H^{\text{pair}}(x_G \cup y_{G^c})
+\sum_{i \in G: y_i \ne *} U^{y_i}(x_i)
+\sum_q
H^{q}(x_{\Lambda^q}|x_{G}) . $$
Here, $ H^{q}(x_{\Lambda^q}|x_{G})
\equiv H^{q}(x_{\Lambda^q}|x_{\partial
(\Lambda^q)^c}) $
and $H^Q_y(x_{\Lambda})$ denotes an `inhomogeneous' Hamiltonian
(1) with $U$ replaced by $U^q$ or $0$
according to whether $y_i = q$ or $y_i =*$.\enddemo
Now, we will {\bf{insert the expressions}} (13), (14)
for Gaussian partition functions
$Z^q(\Lambda^q| x_{(\Lambda^q)^c})$ into
\footnote{
We do not employ here the expression (13')
of $H(\bar x_M| x_{M^c})$ (for the minimising configurations
$\bar x$ in $\Lambda^q$,
under a given $\{\gamma_n\}$).
This will be exploited later (27) in a context
of thick precontours.}
the formula (21).
Say that a path $P$ crosses the set
$\Lambda^q$ if it intersects both $\Lambda^q$ as well
as its complement. Denote such a path by a symbol $P^q$,
and associate to such a path the value $w_{P^q}$ given
by the formula (9), for the quadratic Hamiltonian $H^q$.
\footnote{ Two identical paths $P^q$, $P^{q'}$
will be treated as different ones if $q \ne q'$
in formulas (22).} The forthcoming formulas will
be used for $M \subset \Lambda^q$.
\proclaim{Proposition 3}
Denote by $H^q_e(\bar x_{M}|\bar x_{M^c}) =
H^q(\bar x_{M}| \bar x_{M^c}) - e_q|M|$. Then
$$ Z^q (M|\bar x_{M^c})=
\exp \bigl(-h^q |M| -H^q_e(\bar x_{M}|\bar x_{M^c})
+ \sum_{P^q: P^q \cup M^c \ne \emptyset}
\tilde w_{P^q}^q \bigr)
\tag 22 $$ where $h_q =
e_q + \log \pi -\log \alpha_q -
\sum_{P^q \owns 0} \frac{w_P^q}
{|P^q|}$ is the free energy (10)
and the correction terms
$\tilde w_{\tilde P} = -
\frac{|P^q\cap M|}{2|P^q|} \ w_{P^q}^q$ \
are taken
from (14), for the Hamiltonian $H^q$.
\footnote{
$H^q(q_M| q_{M^c}) = e_q \ |M|$ is the
minimum, over $x_M \in \er^M$, of $H^q( x_{M}| q_{M^c})$.}
\endproclaim
\demo{Proof} It follows immediately from (13) and (14).
\enddemo\remark{Notation} Below, it will be
convenient to write
$ x_{M^c}$ instead of $\bar x_{M^c}$. The quantities
$H^q(\bar x_M| x_{M^c})$ and
$H^q_e(\bar x_M| x_{M^c})$ will then have the same meaning as
with $\bar x_{M^c}$.\endremark
We are now quite close to our goal -- to realise that
the partition functions (21) are (after the substitution
of the expressions (22)) just partition functions
of a suitable `abstract \ps \ model'.
However, the quantities
$ H^q_e(\bar x_{\Lambda^q})| \bar x_{(\Lambda^q)^c})$
and the perturbing terms $\tilde w_P$
obscure the picture. They have to be added
to the `precontour energies'
$E(\gamma) =\log |w_{\gamma}|$ given by (20)
but we have to control the fact
that
$H^q_e(\bar x_{\Lambda^q}| x_{(\Lambda^q)^c})$
is {\bf{not}} additive as the function $\{\gamma_n\}$.
Small corrections to the additivity could be
handled in the same way as the terms $\tilde w_P$
but we may have to deal also with
a possible `strong nonadditiveness' emerging if
the values of spins $x_i$ on the boundary
$\partial (\Lambda^q)^c$ differ very
much
from their `bottom' value $q$.
The remedy is to glue together the precontours
which are `too close' to each other, the meaning of
`closeness' depending also on how much the spins
$x_i$ on the boundary $\{i \in
\partial \cup_i \supp \gamma_n, \dist(i, \Lambda^q) = 1 \}$
differ from $q$:
\definition {Protection zones and thick precontours}
\enddefinition We take the
correlation length of our
quadratic Hamiltonian $H^q$ equal to $1$
(because of the assumed smallness of $\kappa^q$).
Define the
{\bf{protection zone}} $\tilde \Lambda^q$ of the boundary
$\partial (\Lambda^q)^c$ as
the union of balls, with centers in
$ \partial (\Lambda^q)^c$ $\subset \cup_n \supp \gamma_n$
\footnote{
The constant $5 $ is enough `generous'.
This will help us later, when discussing
the Peierls condition.
The exponential decay (15) of
$\bar x$
will guarantee $\varepsilon$ in (23')
of the order $\approx e^{-5}$.}
$$
\ \tilde \Lambda^q = \cup_{i \in
(\partial \Lambda^q)^c} \ B_i^q \ \
\text{where} \ \
B_i^q = \{j \in \Lambda^q: |j-i| < 5 \
( \log ( |x_j -q|/ \delta_q)+1)\}. \tag 23
$$
Denote by $\Lambda^q_{\text{inn}} = \Lambda^q \setminus
\tilde \Lambda^q$ the complement of
the protection
zone $\tilde \Lambda^q$
in $\Lambda^q$.
Let us call $\Lambda^q_{\text{inn}} $ as
the set of all {\bf{inner}} correct (or simply inner) points of
$\Lambda^q$.
Define now thick precontours of $x$
as follows. {\bf{Supports}} of these thick precontours
are defined as connected components of the set
$\Lambda \setminus
\cup_q \Lambda^q_{\text{inn}}$ . Denote them by symbols $G$.
For any such $G$ denote by $G^*$ its {\bf{core}}
which is defined as the union $\cup_n \supp \gamma_n$
of supports of all precontours $\gamma_n$ of $x$ contained in the
`thick' set $G$.
{\bf{Thick precontour}} $\varGamma$ is now defined as a
pair $(y_G, x_{G^*})$ where
$x_{G^*} = \cup_n \gamma_n$ denotes the restriction of $x$ to
$G^*$. Denote them by symbols
\footnote{ {\bf{Contours}}
will be later constructed from thick precontours
and
denoted by symbols $\gb$.} $\varGamma$.
We will also use the notations $G =\supp \varGamma$
and $G^* =\supp^* \varGamma$ (the `core' of $\supp \varGamma$).
Later, we will rewrite (21) in the following,
formally similar to (21) but more convenient form (25).
Notice first that for any thick precontour $\varGamma$ we
have a bound $ |\bar x_i -q| \leq \tilde \varepsilon
$ for any $i \in \partial
(\Lambda^q)^c $, if we take
the configuration $\bar x_{\Lambda^q}$
minimising the quadratic
Hamiltonian $H^q(x_{\Lambda^q}|x_{G^*})$
under a fixed $x_{G^*}$.
In other words,
$$|\bar x_i -y_i | \leq \varepsilon \ \ \ \text{for any} \ \ \
i \in \partial \supp \varGamma \tag 23'$$
where $\tilde
\varepsilon \ \ ( < \delta) $ depends on the choice
of a constant $5$ in the definition of $\varGamma$.
Having any thick precontour $\varGamma =
(G,x_{G^*})$, where $G^* = \cup_n \supp \gamma_n$
(the union is over all precontours
$\gamma_n$ forming $\varGamma$)
and $G = G^* \cup \cup_q \tilde \Lambda^q$
(the last union is over all protection zones $\tilde \Lambda^q$)
we define, using (20), the {\bf{Gibbs factor}} of
$\varGamma$
$$ w_{\varGamma}
=
\prod_n w_{\gamma_n} \prod_q \exp(-H^q(\bar x_{\tilde \Lambda^q}
| \cup_n \gamma_n
\cup q_{\Lambda^q_{\text{inn}}})). \tag 24 $$
where $\tilde \Lambda^q $ is the $q$--th
protection zone
and $\bar x_{\Lambda^q}$ is the
minimising configuration.
\footnote{ Using (23'), we will see that the value
$ H^q(\bar x_{\tilde \Lambda^q} | x_{G^*}
\cup q_{\Lambda^q_{\text{inn}}}) +
e_q \,| \Lambda^q_{\text{inn}}| $ in (24),
with the boundary condition $q$ on
$\partial \Lambda^q_{\text{inn}} = (\partial
\tilde \Lambda^q)^c\setminus G^*$,
is a good approximation
of of $H^q(\bar x_ {\Lambda^q}| x_{G^*})$.}
Here and below we use notations like
$H^q(x_{\Lambda^q }| \gamma)$ instead of
$H^q(x_{\Lambda^q}| (x_{\supp \gamma})$).
\definition{Notation:
Restricted partition functions $Z_{\Cal M}
(\Lambda)$}
In the following, we will introduce a general symbol
$Z_{\Cal M}(\Lambda)$ for partition functions taken over
all configurations satisfying some limitation
$\Cal M$. Such a limitation
will be always expressed in terms of values of $x_i$
in the
incorrect points of a pair
$(x_{\Lambda},y_{\Lambda})$. It can be therefore
equivalently formulated
in the language of (thick) precontours of a pair
$(x_{\Lambda},y_{\Lambda})$. A basic example of such a
limitation $\Cal M$ will be a
condition on the maximal possible {\bf{diameter}}
of the thick precontours of
$(x_{\Lambda},y_{\Lambda})$. \footnote{
Such a limitation will depend also on the
`external colour' of $\varGamma$ and
on $\{x_i, i \in \partial G^*\}$.}
Often we will have $\Cal M = \emptyset $.
\enddefinition
\proclaim{Proposition 4}
Then we have, by (21),(13) and (24), the expression
$$ Z_{\Cal M} (\Lambda) =
\int
\prod_n w_{ \varGamma_{n} } \prod_{q }
Z^q(\Lambda^q|q_{(\Lambda^q)^c}) \
\prod_q
\exp(-H^q_e(\bar x_{\Lambda^q_{\text{inn}}}|\bar x_{\tilde
\Lambda^q}))
\prod_n \ d \varGamma_n^*
\tag 25 $$ where $G =\cup_n \supp \varGamma_n^* $.
See Proposition 3
for the definition of $
H^q_e(\bar x_{\Lambda^q_{\text{inn}}}|\bar
x_{\tilde \Lambda^q})$;\ \
these terms are small and so we have an approximate
relation
$$ Z_{\Cal M} (\Lambda)
\doteq
\int_{\er^G}
\prod_{n} w_{ \varGamma_{n}} \prod_{q}
Z^q(\Lambda^q|q_{(\Lambda^q)^c})
\prod_n \ d \varGamma_n^* . \tag 25'
$$ The symbol
$\prod_n d \, \varGamma^*_n$
means summation and integration over all
possible $y_{\Lambda} = y_{\Lambda}^{\{ \varGamma_n\}}$
and all possible values $x_i
\in \er ,i \in \supp^* \varGamma_n$ where
$\{\varGamma_n \} $ is
an
allowed collection
of thick precontours (each $\varGamma_n$
being written
as $ (y_{\supp \varGamma_n}, x_{\varGamma^*_n})$)
satisfying
$\Cal M$.
\endproclaim
\demo{Proof} The first relation is just a
rearranged expression
(21), taking in account the notation (24) and
the relation, for any decomposition
$M \cup M' \cup M''= \Lambda$,
$ H^{q}(x_{M''}|x_M \cup x_{M'})
+ H^{q}(x_{M'}|x_M \cup q_{M''})
=H^{q}(x_{M^c}|x_M) $.
(Take $M''
=\Lambda^q_{\text{inn}}$, $M'= \tilde \Lambda^q$
and $ M = G \cup \cup_{q' \ne q} \Lambda^{q'}$.)
The second relation follows from (25) and (23'), see
more precisely the definition of a {\bf{contour}} given
below and the relation
(27).
\enddemo
\head 6. Loops and semiloops. Contours
\endhead
\definition{Applying Theorem 0:
Short and long semiloops anchored in $(\Lambda^q)^c$}
\enddefinition
Now we will give another, more detailed
expression (27) of $Z_{\Cal M}(\Lambda)$ which will
be obtained by
inserting
the expressions
(13') and (14) of Theorem 0 into (25).
Before doing so, let us introduce some further terminology.
Let us call the closed oriented
paths $P^q$ as {\bf{loops}},
and oriented paths `from $M^c$
to $M^c$ through $M$' as {\bf{semiloops}}
`anchored' in $M^c$.
While the weights $\tilde w_{P^q}$ of the loops $P^q$ in
(14)
are always small,
so that their contributions can be
studied by the `high temperature expansion'
$\exp(\tilde w_{P^q}) = 1
+ w_{P^q}^*$, the weights $ w_{P^q} \ (x-q)_{s(P^q)}
(x-q)_{e(P^q)}$
corresponding to short semiloops $P^q$
are {\bf{not}} always small. They will be treated
differently:
Let $s(P^q)$ resp. $e(P^q)$ be the starting resp. the
ending
point of the semiloop $P^q$ in $\Lambda^q$ (i.e. of
an oriented path $P^q$ going through $\Lambda^q$
from $s(P^q) \in
\partial (\Lambda^q)^c$ to $e(P^q) \in \partial
(\Lambda^q)^c$).
Say that $P^q$ is {\bf{short}} resp. {\bf{long}}
if any point of $P^q$ has a distance
at most $ 5 \ \log( |x_i -q|/ \delta_q +1)$
from a suitable
point $i$ of the set
$\partial(\Lambda^q)^c$
resp. if
there is a point of $P^q$ having a bigger distance
than $
5 (\log |x_i -q| / \delta_q +1)$ from any point $i$ of
that set. In other words,
short semiloops are precisely those ones
living wholly in the `short' protection
zones $\tilde \Lambda^q$ constructed
in the definition of a `thick' $\varGamma$.
For long semiloops $P^q$, the term
$(x-q)_{s(P^q)}(x-q)_{e(P^q)}$ is
`sufficiently damped'
by the weight $w_{P^q}$, so that the whole term
$ w_{P^q} \ (x-q)_{s(P^q)}(x-q)_{e(P^q)}$
is small.
On the other hand, the sum
over all short semiloops
$S^q$ `going from $(\Lambda^q)^c$ back'
$$ \sum_{S^q} w_{S^q} \ (x-q)_{s(S^q)}(x-q)_{e(S^q)}
= - H^q_e(\bar x_{\tilde \Lambda^q}|x_{\partial (\Lambda^q)^c}
\cup q_{\Lambda^q \setminus \tilde \Lambda^q}) \tag 26$$
gives another expression for the minimum of
the quadratic, relative energy term
$H^q_e( x_{\tilde \Lambda^q}| x_{\partial (\Lambda^q)^c}
\cup q_{ \Lambda^q_{\text{inn}} })$
defined in (22).
Here,
$\tilde \Lambda^q = \Lambda^q \setminus
\Lambda^q_{\text{inn}}$ is the
protection zone i.e. the collection of all points of
$\Lambda^q$ whose distance to $\partial (\Lambda^q)^c$
is `short' i.e. smaller than $ 5
(\log(|x_i -q| / \delta_q +1)$
for some $i$.
Recall that the boundary condition
$x_{\partial (\Lambda^q)^c}$ is determined
by the collection of the values of precontours
$ \gamma_{n,j}$ of the system.
Using (22) and separating long and short semiloops
we rewrite (25) as follows:
\proclaim{Proposition 5}
Denote by $\bar x_{\Lambda^q}$ the configuration
minimising $H^q ( \, \cdot \, |x_{\partial (\Lambda^q)^c})$.
Then
$$ Z_{\Cal M} (\Lambda) = \int \
\prod_n w_{\varGamma_n} \ \prod_q
\exp(- h_q |\Lambda^q|)
\prod_q \exp( \Delta^q(\{\varGamma_n\}))
\prod_n d \, \varGamma^*_{n} \
\ \ \ \ \text{where} \tag 27
$$ $$ \Delta(\{\varGamma_n\}) =
\sum_{L^q \in \Cal L^q}
w_{L^q} ( x-q)_{s(L)} (x-q)_{e(L)} \ +
\sum_{P^q \in \Cal P^q}
\tilde w_{P^q}
\ \ ; \ \tilde w_{P^q} =
\frac{|P^q \cap (\Lambda^q)^c|}{|P^q|}
w_{P^q}.
$$ We write $\int g(\{\varGamma_n\})\prod_n
d \varGamma^*_n$
instead of a more precise
$\sum_{y_{\Lambda}} \int g(\{\varGamma_n\})\prod_{n,j}
d \, x_{G_{n,j}}$ here. The sum is over
all possible colorings $y_{\Lambda} =
y_{\Lambda}^{\{\varGamma_n\}}$ (for some allowed
collection
$\{\varGamma_{n}\} = \{ \gamma_{n,j}\} $) and the integral
is over
all possible values
$ \{x_{G_{n,j}}, G_{n,j} = \supp \gamma_{n,j}\}$
of the `core' $\{\varGamma^*_n\}$,
satisfying $\Cal M$.
The symbol $\Cal L^q$ \
denotes
the collection of all
{\bf{long semiloops}}
$L^q$ going from $\partial (\Lambda^q)^c$
back (through $\Lambda^q$) and $\Cal P^q$ denotes
the collection
of all {\bf{loops}} $P^q$ intersecting both
$\Lambda^q$ and $(\Lambda^q)^c$.
\endproclaim
\demo{Proof} See (25) and (22). The term
$H^q_e(\bar x_{ \Lambda^q}|x_{\partial (\Lambda^q)^c}) $,
extracted from the partition functions $Z^q$ in (22)
is now decomposed into the `short'
sum (26) (added to the energy of $\varGamma_n$, because
its terms may be dangerously large) and the
remainder
$$ H^q_e(\bar x_{\Lambda^q_{\text{inn}}}|
\bar x_{\tilde \Lambda^q})
= - \sum_{L^q \in \Cal L^q}
w_{L^q} ( x-q)_{s(L^q)},(x-q)_{e(L^q)} \tag 26' $$
which is a sum of small terms indexed by `long' semiloops $L^q$.
(We were using the conditioned Hamiltonian of the type
(12) everywhere;
notice that then the condition $x_i = q,
\ i \in \Lambda^q_{\text{inn}}$
has the same effect as the empty boundary condition.)
Thus, (27) appears when we substitute (26') and (14)
into (25).
\enddemo
For any long semiloop $L^q$ going `from $(\Lambda^q)^c$
back' define now the quantity
$$ w^*_{L^q} \equiv w^*_{L^q}((x-q)_{s(L^q)},(x-q)_{e(L^q)})
= \exp(w_{L^q} \ (x-q)_{s(L^q)}(x-q)_{e(L^q)}) -1 . \tag 28 $$
Let us use the notations $P^q$ resp. $L^q$ for loops resp.
long semiloops also in the following definition, which
summarises
all the previous constructions of the paper.
\remark{Note} The terms
$ w_{S^q} \ (x-q)_{s({S^q})}(x-q)_{e(S^{q})}$ for
short semiloops $S^q$ were already summed in
(26). They are
added to the
energy of the precontours.\endremark
\definition{ Contours} \enddefinition
They will be defined as suitable triples $
\gb= (\Cal A, \Cal C,
\Cal L) $
where $\Cal A$ is an
allowed collection $\{\varGamma_n\}$
of {\bf{thick precontours}}, $\Cal C = \cup_q \Cal C^q$ is
some
collection
of {\bf{ loops}} (oriented paths, `cycles') in $\Lambda$
such that each $P^q$ from the system $\Cal C^q$ intersects
both the sets
$\Lambda^q$ and $(\Lambda^q)^c$, and
$\Cal L = \cup_q
\Cal L^q$
where $\Cal L^q$ is a collection of
{\bf{long semiloops}} $L^q$
in $\Lambda^q$, which start and end
in $(\Lambda^q)^c$. The condition on the whole system
$\gb = (\Cal A, \Cal C,
\Cal L) $
is that we require the {\bf{connectedness}} of the
support
$$\supp \gb
:= ( \cup_{\varGamma \in \Cal A} \supp \varGamma )\cup
(\cup_{P \in \Cal C}
\supp P) \cup (\cup_{L \in \Cal L} \supp L )
. $$
Contours will be denoted
by symbols $\gb$. We will use also another notation
$\supp^* \gb = \cup_{\varGamma_n} \supp^* \varGamma_n$
for the support of the {\bf{core}} $\gb^*$ of $\gb$.
We associate to each contour $\gb$
its {\bf{Gibbs weight}} $w_{\gb}$ determined,
with the help of $w_{\varGamma_n}$, as follows. Put
$$ w_{\gb} = \prod_{\varGamma_n \in \Cal A}
w_{\varGamma_n}\ \
\prod_{q \in Q} \ \
\prod_{P^q \in \Cal C^q}
w^*_{P^q} \ \prod_{L^q \in \Cal L^q} w^*_{L^q} \tag 29$$
where the weights $w_{\varGamma} $ are given by (24)
(see also (20), (26) for more details). The weights
$ w^*_{L^q}$ are given by (28) and the weights
$w^*_{P^q}$ by (compare also (14))
$$ w^*_{P^q} = \exp(\tilde w_{P^q}) - 1 =
\exp(\frac{|P\cap \Lambda^q|}{2 |P|} w_{P^q}) - 1. \tag 30
$$
In later investigations,
we will write $w_{\gb}$ in
the exponential form
$$ \exp(-E(\gb)) :=
|w_{\gb}| = \prod_{
\varGamma_n \in \Cal A }|w_{\varGamma_n}|
\ \ \prod_{P^q \in \Cal C^q}
|w^*_{P^q}| \ \prod_{L^q \in \Cal L^q} |w^*_{L^q}|
. \tag 29' $$
\head 7. Frames. The Abstract Pirogov Sinai
Model \endhead
The following modification of the notion of a contour
-- which will be used systematically below -- is now
only a `cosmetic' one. Namely, we
will see in (33) that the values $x_i,\, i \in
\supp \gb^*$ can be integrated out
in the expressions (27) of partition functions
$Z_{\Cal M}(\Lambda)$.
\footnote{In (21) and (25), the values of
$ \gb^*$
were needed also as an information on {\bf{boundary
conditions}}
for Gaussian partition functions
$Z^q(\Lambda^q| x_{(\Lambda^q)^c})$.
In (27), the only objects
depending on the {\bf{values}} of $\gamma_{n,j}$ are
$\supp \varGamma_n$ and also the terms
$ w^*_{L^q}(( x-q)_{s(L)},(x-q)_{e(L)})$.}
This suggests to apply the following
factorisation of contours $\gb$, by
`forgetting' all the
values of $\gb^*$ (but not the colours of $\gb^*$):
By a {\bf{frame}}
of a contour $\gb$ we will mean a pair $\Cal F =
(\supp \Cal F , y_{\Cal F} )$ where
$\supp \Cal F = \supp \gb$
and $y_{\Cal F}$ is the colour of
$\Cal F$: $ y_{\Cal F} = \{y_i,
i \in \supp \gb\} \in (Q\cup*)^{\supp \gb}$.
Put
$$
w_{ \Cal F} =
w_{(\supp \Cal F,y_{\Cal F})} =
\int_{\gb \in \Cal F} w_{\gb} \
d \gb^*
\tag 31$$
where the integral is over all the values of the cores $\gb^*$
of contours $\gb$
having the same frame $\Cal F=
(\supp \Cal F , y_{\Cal F})$.
We introduce also the notation $\supp^* \Cal F = \supp^* \gb$.
\remark{Note}
The paths $P^q, L^q$ live in the Gaussian model
given by $H^q$; they have a constant colour $q$. On the
other
hand, the `inner' colour of a (thick) precontour can
be different from the outer one. Frames still
carry the information
about a possible change of a colour inside $\gb$.
The values $w_{ \Cal F}$
depend also on these colours.
\endremark
\definition {Signed frames}
\enddefinition
Below. we will work only with the frames.
Namely, these objects will be the `contours'
of our new, `abstract \ps \ model'.
Notice that neither $w_{\gb}$ nor $w_{ \Cal F}$
have to be positive.
Introduce a concept of a {\bf{sign}}
of a frame $ \Cal F$ $$
w_{ \Cal F} = \sign{ \Cal F} \
\exp(-E( \Cal F)) \tag 32 $$
where $\sign( \Cal F) = \pm$.
The accompanying quantity $E(\Cal F)$ is the
{\bf{energy}} of
$\Cal F$.
See the discussion below (33).
We mention in advance that the value of $\sign \Cal F$
has to be kept also for the
{\bf{contour functional}} $F(\Cal F)$ of $ \Cal F$,
an important concept of
the discrete spin \ps \ theory (see \cite{S}, \cite{PS},
\cite{ZR})
used to study (33).
\remark{Note}
Most papers
from \ps \ theory do not
consider signed contours.
However, it is not difficult to adapt e.g.
\cite{ZR} for such a case.
One could introduce (and this is a much more serious problem)
even a more
general notion of a {\bf{complex}} modulus
$\sign \Cal F \in \{ z: |z| = 1\}$ of a
contour (frame) \ as an attribute
of $\Cal F$ appearing always together with
the (real) energy $E(\Cal F)$. (Thus introducing a
general concept of a {\bf{complex Hamiltonian}}
for abstract \ps \ models.)
This problem deserves a separate paper.
The general complex
modulus $\sign(\Cal F)$ will
change its value appropriately when
going from $E(\Cal F)$ to the contour functional $F(\Cal F)$.
What is now lost is
the
{\bf{probabilistic}} interpretation of some` results;
however the definition of quantities like $h_q$
can be carried also to this complex case.
Such an approach would yield appropriate
piecewise analyticity results
for the phase diagram. We plan to write
a new variant of the paper \cite{ZA}
where these questions would be handled in
the new language of \cite{ZR}. \endremark
Let us finish the promised transcription,
of our continuous spin model,
to an `abstract \ps \ model'.
>From (27), (29)
we have the following conclusion:
\proclaim{Theorem 1}
Any partition function
$Z_{\Cal M} (\Lambda)$ can be expressed as
$$Z_{\Cal M}(\Lambda)=
\int
\prod_n w_{\gb_n}\prod_{q \in Q}
\exp(-h^q |\Lambda^q|) )
\ d \{\gb_n^*\} \ =
\sum_{\{\Cal F_n\}} \prod_n w_{\Cal F_n}
\prod_{q \in Q}
\exp (-h^q |\Lambda^q|)= $$ $$ = \
\sum_{\{ \Cal F_n \}} \ \prod_n \ \sign
\Cal F_n \,
\exp(- E( \Cal F_n)) )
\prod_q \ \exp (-h^q \,|\Lambda^q|)
\tag 33 $$
where $w_{\gb}$ resp.
$w_{\bar \Cal F}$ are given as above in (29) resp. (31).
The integral resp. sum in (33) is
over all allowed collections of
contours $\gb_n$ resp. frames $\Cal F_n$ obeying $\Cal M$.
\endproclaim
\demo{Proof} This follows directly from
Proposition 5
and from the definition of contours (first relation)
and frames (second relation). Then we use the notations
(32).
\enddemo
\remark{Notes} 1.
We note that the true {\bf{energy barrier}}
of a frame
$\Cal F$ (with respect to the constants $q$ on $\Lambda^q$)
is not $E( \Cal F)$ but
$ E(\Cal F) +h |\supp^* \Cal F|$ \
where $h$ is something like $h = \min h_q$.
Namely, take in mind the `holes' $\supp^* \Cal F_n$
in (33) i.e. the relation
$$ \Lambda = \cup_q \Lambda^q
\cup_n \supp \Cal F^*_n . $$
In usual examples, the differences between various
$h_q$ are small and one can assume that $h_q \equiv 0$
when discussing the validity of the Peierls condition.
2. Having established the Peierls condition, the technology
of the discrete spin \ps \ theory (\cite{ZR})
can be used. The limitation $\Cal M$ can have a
meaning
that the `external' thick precontours of an allowed
configuration do not have an `astronomical' size;
being `considerably smaller' than
the size of $\Lambda$. Such an assumption
is also useful if we want to have
an intuitively obvious definition of
`external contours' on a torus. However, the choice
of a selected point `$\infty$' in $\Lambda$, see Theorem 3,
is formally simpler.
Then we get an analogy of usual
$q$--diluted partition functions which are now more precisely
taken over configurations with
`nonastronomical' contours only (resp. taken over
configurations
having the prescribed value $y_{\infty} = q$).
The limitation $\Cal M$ can have also some more subtle meaning
namely the `metastability' of any considered configuration.
To express such a property
(\cite{Z}, \cite{ZR}) of the `abstract \ps \ model'
in the language of original configurations
$x_{\Lambda} \in \er^{\Lambda}$
(and `colorings' $y$) is slightly
cumbersome. Fortunately, the problems of an
interpretation of the partition functions
(33))
appear for `nonstable' values of $q$ only (and
for very large volumes);
otherwise we usually have
{\bf{no}} limitation $\Cal M$ in the expressions (33).
\endremark
\head {8. The Peierls Condition}
\endhead
It remains to check the
Peierls condition for the energies $E( \Cal F)$
in (33).
Then we will really have an `abstract \ps \ model'
and the analysis of the events
which can be formulated in terms of the partition functions
(33) will be possible.
See the forthcoming
section for some interpretation. With regard to
(31), (29') we establish below in (38)
suitable upper bounds for
the activities $ |w_{\gb}|$ and
$|w_{\bar \Cal F}|$:
\bigskip
\definition {Investigation of the
structure of the contour and frame energy }
\enddefinition
We investigate below the energy $E(\Cal F)$ of a frame
$\Cal F$ of a contour
$\gb = (\Cal A, \Cal C, \Cal L)$
where $\Cal A = \{\varGamma_i\} $
is a collection of thick
precontours and $ \Cal C =\{\Cal C^q\}$ and $ \Cal L=
\{ \Cal L^q \}$ are systems
of loops and long semiloops. Write the Gibbs factor (29)
$w_{\gb} $ as
$$ w_{\gb} = \prod_{i} w_{\varGamma_{i}}
\prod_{q \in Q}
\exp (
\prod_{P^q \in \Cal C^q}
\prod_{L^q \in \Cal L^q} w^*_{P^q }\ w^*_{L^q}
= \tag 34$$ $$ =
\prod_{i,j} w_{\gamma_{i,j}}
\prod_q \exp (-H^q(\bar x_{\tilde \Lambda^q}
|\cup_{i,j} \gamma_{i,j} \cup q_{\Lambda^q_{\text{inn}}})
\prod_{P^q \in \Cal C^q}\prod_{ L^q \in \Cal L^q}
w^*_{P^q} \ w^*_{L^q} $$
where $\{\gamma_{i,j}\}$ is the collection
of precontours contained in the thick
precontour $\varGamma_i$ and
$\tilde \Lambda^q$ is the `$q$--short neighborhood'
(protection zone)
of $\partial (\Lambda^q)^c$ introduced
in the definition (23) of
thick precontours. The configuration
$\bar x_{\tilde \Lambda}$ optimizes the value of
$H^q(x_{\Lambda^q}|
\cup_{i,j} \gamma_{i,j})$.
The quantities $w^*_L$ resp. $w_P^*$ were defined in
(28) resp. (29').
The formula (34) follows from (29) and (24),
and it can be also written, using
auxiliary notations
$E(\gb) = - \log |w_{\gb}|$, $E(P) = -\log |w^{*}_P|$ ,
$E(L) = -\log |w^*_L|$,
in the following, more compact form
($y$ denotes the colour
induced by $\gb$)
$$
\exp(-E(\gb)) =
\exp(-H_y(\bar x_G|y_{G^c} ) \prod_{P^q \in \Cal C^q}
\exp(-E(P^q)) \prod_{L^q \in \Cal L^q}
\exp(-E(L^q)) . \tag 35$$
Here, $G$ is defined as
$G = G^* \cup \tilde \Lambda$ with the core
$G^* := \cup_{i,j} \supp \gamma_{i,j}$
(union over all precontours
of $\gb$) and
with $\tilde \Lambda$ being
given as follows:
$\tilde \Lambda = \cup_q \tilde \Lambda^q$ where
$\tilde \Lambda^q$ denotes
the $q$--th protection zone of $ G^* $. Alternatively,
$\tilde \Lambda^q$
is equal to
the union of `truncated' (excluding start- and endpoints)
supports
of all short semiloops in $\Lambda^q$
going from $\partial G^*$
back to $\partial G^*$ through $\Lambda^q$.
The configuration $\bar x$ and the
Hamiltonian $H_y(\bar x_G|y_{G^c})$ are defined as follows:
\roster
\item The configuration $\bar x_G $ is chosen to
be equal to the values of $\cup_{i,j}
\gamma_{i,j}$ on $G^*$. It is extended
to the set $G = G^* \cup \cup_q \tilde \Lambda^q$
in such a way that it
minimises, in each $\tilde \Lambda^q$,
the quadratic Hamiltonian
$H^q(x_{\tilde \Lambda^q}|x_{G^*}
\cup q_{ \Lambda^q_{\text{inn}}})$.
\footnote{This new $\bar x$ is `slightly improved' (w.r. to
the
boundary condition $q_{ \Lambda^q_{\text{inn}}}$)
compared to (34).} \item
The pair Hamiltonian
$ H^{\text{pair}}(\bar x_G|y_{G^c}) $ is used in (35).
For any $k \in G$ we add
thepotential $U^{y_k}(\bar x_k)$
where $y_{\Lambda} = \{y_k,\ k \in \Lambda\}$
denotes the
colour induced by $\{\gamma_{i,j}\}$. In particular
for any
$k \in G \setminus G^*$ which is a
$q$--correct point of $y_{\Lambda}$
we add the potential
$U^q(\bar x_i)$ when defining $H_y(\bar x_G|y_{G^c})$. \endroster Thus,
to check
the validity of the
Peierls condition we have to analyze the behaviour
of the three types of `energies': the `physical'
quantity
$H_y(\bar x_G|y_{G^c})$
and also the `artificial' quantities $E(P), P \in
\Cal C$ and $E(L), L \in \Cal L$.
While it is rather straightforward to check the Peierls
condition
for the latter two terms (for small constants $\kappa_q$, and
with slight modifications needed for the long semiloops $L^q$,
compared to the case of loops $P^q$, see
(45)) the investigation of $H(\bar x_G|y_{G^c})$
requires more care.
We prove the
required bounds
for $H_y(\bar x_G|y_{G^c})$ only in the special case
$R =1$. The notation $\tilde H_y(x_G)$ used below
for $
H_y(x_G| y_{G^c})$ suggests that now it is more
convenient to think in terms of interactions $\sum_{i,j}
b_{i-j} (x_i -x_j)^2$
and relative Hamiltonians $\tilde H^q(x_M|x_{M^c})$ instead of
quadratic forms
$\sum_{i,j} A^q_{i,j}(x_i-q)(x_j-q)$ and (12).
\definition{Notations} Say that $i\in \Lambda $ is a
{\bf{$q,*$--correct
point}} of a coloring $y_{\Lambda}$ if the values $y_j, \
|j-i| \leq 1$ are equal either to $q$,
at least for one such $j$,
or to $*$. Say that $i$ is a {\bf{wild}} point of
$y_{\Lambda}$
if it is $q,*$--correct for {\bf{no}} $q \in Q\cup\{*\}$.
Decompose further the collection $\Lambda^{q,*}$ of
all $q,*$--correct points of $y_{\Lambda}$
into $\Lambda^{q,*}_q \cup
\Lambda^{q,*}_*$, according to whether
$y_i = q $ or $y_i =*$.
Finally, say that
$x_i$ where $ i \in \Lambda^{q,*}_*$ is
{\bf{ $q$--like}} if $x_i \in
(q', q'') $ (where $q' < q < q''$ are nearest neighbors
from $Q \cup \{\infty\} \cup\{-\infty\}$) otherwise
`$q$--redundant'.
\enddefinition
\proclaim{Theorem 2}
Consider the model (1), (2), (3) with $e_q \equiv 0$.
Let a connected set $G$ be given,
$G = \cup_{q \in Q} \ G^q \cup G^*$
such that $G^*$ is the set of all
incorrect points of
some $y =y_{\Lambda} \in (Q^*)^{\Lambda}$
and all points of $G^q$ are $q$--correct.
Let
$x_{G^*}$ be a configuration on
$G^*$.
Decompose $G^*$ into
`wild' resp. `$*,*$--correct' resp.
`$q,*$--correct' parts
$$ G^* = G^{w} \cup G^{*,*} \cup \cup_{q \in Q}
G^{q,*}\ , \ \ G^{q,*} = G^{q,*}_q \cup G^{q,*}_*.
\tag 36$$ Assume that the union of `protecting
balls' $B_j^q$ with
centers in $G^{q,*}_*$
and with radiuses $ 5
(\log ( |x_j -q|/\delta_q)+1) $
covers the set $G^q$, for each $q \in Q$.
Extend $x_{G^*}$ in an optimal way to $G$ so that
the minimal value of the expression
$$ \tilde H_y(x_G) = \sum_{\{i,j\} \subset G \cup
\partial (G^c)}
b_{i-j} (x_i -x_j)^2 + \sum_{q \in Q}
\sum_{j \in G^q} U^q(x_j)
+ \sum_{j \in G^*} U^{y_i}(x_j) \tag 37$$
is attained,
under the boundary condition
$x_{\partial G^c} =y_{\partial G^c}$ (compare (35).
\footnote{ Namely, for $\partial G^q \setminus
\partial (G^*)^c$ $ = \cup_q \partial
\Lambda^q_{\text{inn}} $ we have assumed that
$x_{\partial G^c} = y_{\partial G^c} $.}
Fix the value $y_i$ in a selected point (`$\infty$')
of $\Lambda$ and
take the sum and integral over all possible
$(y_{\Lambda},x_{G^*})$ giving the same `incorrect' set
$G^*$ and `protection zones' $G^q$.
Then the assumptions (17 B), (17C), (17E) imply a bound
(with
$\tilde \varepsilon \asymp \varepsilon_{(17B)}+\exp(-\alpha)$)
$$\sum_{y_{\Lambda}} \int
\exp(-\tilde H_y(x_G)) \ d x_{G^*} \leq
\tilde \varepsilon^{ \ |G| }.
\tag 38 $$
\endproclaim
\demo{Proof} One has to understand the
contributions of $G^w$, $G^{q,*} $ and $ G^{*,*}$ to the energy
(37). The idea is that for $G^* = G^{q,*} \cup G^{*,*}$,
the sum of $U^*(x_i)$ over $ i \in G^{q,*}_*$
gives sufficient lower bound for
the energy (37) if all $x_i$ are $q$--like.
If $G^{w} \ne \emptyset$, another big energy term
comes from the `wild' quadratic terms in (37), and we
start with the estimate of the quadratic part of (37):
For $x_{G^c} = y_{G^c}$ write (37) as
$$ \tilde H_y(x_G) =
\tilde H_y^{Q}(x_G) + U^*_*(x_{G^*}) +
U^*_w(x_{G^*})
\ \text{where} \
U^*_*(G^*) = \sum_{i \in G^* \setminus G^w}^{y_i =*} U^*(x_i),
\tag 39 $$ $$
U^*_w(x_{G^*}) = \sum_{i \in G^w}^{y_i =*} U^*(x_i)
\ \ \text{and} \ \ \tilde H_y^{Q}(x_{G}) =
\sum_{i \in G: y_i \ne *} U^{y_i}(x_i) \ +
\sum_{\{i,j\} \nsubseteq G^c}
b_{i-j} (x_i - x_j)^2 \
. $$ Assume for
clarity of the argument given below
that $y_i \ne *$ for each $i \in G^w$. (Then
$U^*_w \equiv 0$.) The generalization
to a general
$y_{G^w} $ is not difficult. If
$ y_k = *$ for some $k \in G^w$ (e.g. if
there is a jump from $i \in G^{q,*}_q$
to
$j \in G^{q',*}_{q'}$ with $q' \ne q$ and
with $|k-i|=$ $ |k-j| =1$) then
the
term $U^*_w(x_{G^*})$ must be also employed
in these estimates.
(It is helpful --and possible--
to imagine that
$H^{\text{pair}}(x_{\Lambda})
= \sum_{|i-j|\leq 2} (x_i-x_j)^2$ i.e. to employ
`artificial next nearest neighbour interactions'
in such a case.)
For any
$q,q' \in Q$, $q \ne q'$ we have a bound, with
some $0< \gamma < 1$ and $C \asymp 1/2\nu$
$$ \alpha (x_i -x_j)^2 + 1/2\nu \ ( U^q(x_i) + U^{q'}(x_j))
>\gamma (q - q')^2 +C(x_i -q- x_j + q')^2 .\tag 40$$
Put $\ G' = G\setminus G^{w} $.
Using the notation (37) also for $G^w$ decompose
$\tilde H_y^{Q}(x_{G})$ as
$$ \tilde H_y^{Q}(x_{G})
= \tilde H^Q_y(x_{G'} |
x_{(G')^c})\ +\tilde H_y^Q(x_{G^{w}}) \ ;
\ \tilde H_y^Q(x_{G^{w}}) = \gamma \tilde H_y^Q( y_{G^{w}})
+ \phi_y(x_{G^w})
$$ (take $(q-*)^2 \equiv 0$) where
$\tilde H^Q(x_M|x_{M^c}) =
\sum_{\{i,j\} \nsubseteq M^c}
b_{i-j} (x_i -x_j)^2 + \sum_{i \in M} U^{y_i}(x_i)$.
The remainder $ \phi_y(x_{G^w}) \equiv
\tilde H_y^Q(x_{G^w}) - \gamma \tilde H_y^{Q}( y_{G^w})$
is quickly growing for $\gamma <1$.
Notice that
$\tilde H_y^Q(y_{G^{w}}) \equiv \tilde H_y^Q(y_{G^{w}}\cup
y_{(G^w)^c})$ and $\tilde H_y^{Q}(y_{\Lambda})=
\tilde H_y^{\text{pair}}(y_{\Lambda})$ because $e_q \equiv 0$.
By (40), $\phi_{y}(x_{G^w})
> C |x_{G^w} -\bar y_{G^w}|^2 $
for a suitable $\gamma < 1$ and $ C>0$. Thus, write
$$
\int \exp (- \tilde H_y^{Q}(x_{G^w})) d x_{G^w}
=
\exp(-\gamma \tilde H_y^{Q}(y_{G}))
\int \exp(-\phi_{y}(x_{G^{w}})) \ d x_{G^{w}}. \tag 41 $$
$ \exp(C'|G^w|)$. Summing
over all possible `wild' $y_{G^w}$
yields a term of the order
$ \exp(-\gamma \alpha \delta^2 |{G^{w}}|)$. Namely,
$\exp(- \gamma \tilde H_y^{Q}
(y_{G})) \leq
\exp(- \gamma \alpha |\delta|^2 |G^w|) $ by (17C).
Therefore,
the sum over $y_{G^w}$ of the
integrals (41) is bounded as
$ \varepsilon^{|G^w|}$
with $\lim_{\alpha \to \infty} \varepsilon = 0$.
So we may conclude:
Having suitable Peierls
bounds for the case $G' = G$
we can get them from (17C) also for
a general $G^w \ne \emptyset$, if $\alpha$ is
enough large.
\enddemo
\definition{ Peierls bounds for the case $G^{w} = \emptyset$
and $G^{*,*} = \emptyset$}
\enddefinition
We are essentially left in the case when
$ G^* = G^{q,*} $ for some $q$. Estimate, by (39)
$$
\exp (
- \tilde H^Q_y(x_{G}) -U^*(x_{G^*})) \ \text{where} \ \
\tilde H^Q_y(x_G) = \tilde H^{\text{pair}}(x_G) + \sum_{y_i =q}
U^q(x_i). \tag 42 $$
The smallness of (38) will now be
obtained with the help of (17B)
and the fact that the points with $y_i =*$ are
`dense' in $G^*$. The difficulty is that
$|G^q|$ may be much bigger than
$|G^*|$.
Denote by $N$ a minimal possible collection of
points $i \in G^*$
`responsible for $G^q$'
such that
$y_i =*$
for each $i \in N$ and the union of all
$$B_i^q = \{j: |j-i| \leq 5 (\log(|x_i -q| / \delta_q)+1)\}
$$
covers $G$ : $ G
\subset \cup_{i \in N} \ \ B_i^q$.
Say that $N$ is a `net' of $G^*$.
Fix a coloring $y$, allowing given sets $G^q$
and $G^*$. Consider the integral
$\int_{(N)}
\exp(-\tilde H_y(x_G)) \ d x_{G^*}$ over all $x_{G^*}$
allowing the choice
$ N$ of a net of $G^q$.
Assume that $G^{*,*} =\emptyset$ for a simplicity.
Then
$$
\int_{(N)}
\exp( - \tilde H_y(x_{G}) ) \ d x_{G^*} = \int_{(N)}
\exp (- U^*(x_{G^*}) - H^Q_y(x_G))\ d x_{G^*} \leq \tag 43
$$ $$\leq
C^{|G^*|} \int_{(N)}
\exp (- U^*(x_{G^*}) )\ d x_{N} \leq
\prod_{i \in N} (\sum_{B_i^q}
(C \varepsilon)^{(\diam B_i^q)^{\nu}})
\ \leq \ \prod_{i\in N}
( \sum_{B^q_i}(C'\varepsilon)^{ |B_i^q|})
$$ which is $ \leq \ (C''\varepsilon)^{|G |} $
if we sum over all possible $B^q_i$.
Really, take a function, e.g.
$$
\tilde U^*(x_i) = U^*(x_i) +
\frac{1}{2\nu}((x_i -x_j)^2 + U^q(x_j))
\ \ ; \ \ i \in G^{q,*}_*
\tag 44$$
with $|j-i| = 1, \ j =q$,
to control also the `redundant' case
$x_i \in (q',q'')^c$. Then $\tilde U^*(x_i)$
grows quickly in $(q',q'')^c$ and so we get, by
(17B), the second bound in (43).
The factor
$C^{|G^*|}$ bounds the integral
$\int \exp(- \tilde H^Q_y(x_G)) d x_G^*$. Summing
over all possible $N$
adds another factor
$2^{|G^*|}$ not changing the nature of this estimate.
Finally, the region $G^{*,*} \ne \emptyset$
is controlled
by using the fact that $U^*$ is `uniformly large'
by (17B). Here, again, the quadratic term
$\tilde H^Q_y(x_{G^{*,*}}|x_{(G^{*,*})^c})$
helps to control (using (17C))
the integral over $x_{G^{*,*}}$.
This concludes the proof of Theorem 2.
We need finally also some {\bf{estimates of the terms}}
$w_P^*$ {\bf{and}} $w^*_L$.
For long semiloops $L^q$, the term
$ (x-q)_{s(L^q)}(x-q)_{e(L^q)}$ has a much smaller logarithm
than $(w_L^q)^{-1}$ and so the estimates of the
terms
$ w_{L^q} \ (x-q)_{s( L^q)}(x-q)_{e(L^q)}) $
and $w^*_{L^q}$ can be done
in a similar way as for the
(configuration independent) terms $w_{P^q}$ and $w^*_{P^q}$.
We need crucially the assumption that all
$\kappa^q$ from (5)
are small.
Then, writing $\kappa^q = \exp(-K^q)$ estimate
the
sum over all paths $P^q \owns i$ of a length $n$ by
$$ \sum_{P^q \owns i: |P^q| = n} |w_{P^q}|
\leq \exp((C- K^q) n) \tag 45$$
which gives the necessary complement to (38)
for $\kappa^q$ small. We conclude:
\proclaim{Corollary}
For any collection $\{ \Cal F \}$ of frames with a fixed
colour at a given point of $\Lambda$ and
with a fixed support $\supp \Cal F$,
$$\sum_{\Cal F} \exp(-E( \Cal F))
\leq \exp( -\tau |\supp \Cal F|)
\ \ \text{where} \
\tau = C \min \{- \log \tilde \varepsilon,
-\log \kappa \}.
\tag 46 $$
\endproclaim
\head {9. Conclusion. Example.}
\endhead
Thus we have an
`abstract \ps \ model'
whose `contours' (frames) satisfy (46).
Such models are studied systematically
e.g. in \cite{ZR} (for the case $\sign \gb \equiv 1$ only)
and so we can use the available technology here,
to expand the partition function (33).
What is needed is also a suitable {\bf{interpretation}}
of these expansions, in terms of the
original continuous spin model.
This is rather straightforward for {\bf{stable}} values
$q \in Q$. Stability means the existence of
the `$q$--like' Gibbs state of the discrete spin ensemble
on $\zv$ (constructed for the model of Theorem 1).
For a more detailed information on this crucial notion
see \cite{Z} and \cite{ZR}. It is essential
that
this is equivalent also to the `stability' of $q$ in
the {\bf{continuum spin}} model (1)
i.e. to the existence of a Gibbs
state on $\er^{\zv}$
whose almost all configurations have values `mostly
from' the $q$--th well of $U$.
More precisely we have
the following result, formulated for any torus
and for the following special ensemble,
corresponding to the choice of a `$q$--like
boundary condition in an infinity':
Select a point in $\Lambda$,
denote it by $\infty$ and consider the modified
Hamiltonian $H^q_{\infty}(x_{\Lambda})$ defined by
(1) but with the term $U(x_{\infty} )$ replaced by the
{\bf{quadratic term}} $U^q(x_{\infty})$; formally
$ H^q_{\infty}(x) = H(x) + U^q(x_{\infty}) -U(x_{\infty})
$. \
Such a choice of
a {\bf{fuzzy boundary condition $q$}}
defined in terms of a modified Hamiltonian
$H^q_{\infty}$will be convenient here,
and one has to supplement it by a convenient expression
of the idea that `$x_i$ resides in the $q$-- th well'
(to characterize the behaviour of a typical configuration
in the $q$--th Gibbs state).
Consider the following `smooth indicator of being
outside of the $q$--th well'
namely
$$ f(x_i) = \exp(U(x_i)) \bigl(
\sum_{q' \in Q \setminus \{q\}} \exp(- U^{q'}(x_i)) +
w(x_i)\bigr)
= 1 -\frac{\exp(-U^q(x_i))}{\exp( -U(x ))} . \tag 47 $$
\proclaim{Theorem 3} Let $q$ be stable.
Then
the expectation of the indicator (47)
is expressed by the formula,
where $\mu^q$
is the Gibbs measure on $\er^{\Lambda}$
corresponding to $H^q_{\infty}$
$$ \int f(x_i) d \mu^q (x_i) =
\ \frac{Z^q_{\infty}(\Lambda| *)}
{Z^q_{\infty}(\Lambda)}
=
1- \ \frac{Z^q_{\infty}(\Lambda| \dagger)}
{Z^q_{\infty}(\Lambda)} .
\tag 48$$
The partition functions
$Z^q_{\infty}(\Lambda)$ and
$Z^q_{\infty}(\Lambda|*)$
resp. $Z^q_{\infty}(\Lambda| \dagger)$ are
of the type (18), given as
integrals/sums over all
$(x_{\Lambda},y_{\Lambda})$ satisfying
the additional condition
$y_{\infty} = q$ and also, for the
numerators,
$y_i \ne q$ (case $*$) resp. $y_i = q$ (case $\dagger$).
They are expressed as partition functions
(33), $\Cal M =\emptyset$ and can be expanded like
$$ \frac{Z^q_{\infty}(\Lambda| \dagger)}{Z^q_{\infty}(\Lambda)}
= \exp (\sum_{\Cal C}^{\dagger} n_{\Cal C} w_{\Cal C})
\approx 1 - \omega \tag 49$$
with $\omega = \exp(-C \tau)$. The sum
$\sum^{\dagger}_{\Cal C}n_{\Cal C} w_{\Cal C}$ is taken over all
clusters $\Cal C$
of frames $\Cal F$ {\bf{not touching}}
the point $\infty$
but {\bf{touching}} the point $i$ (in such a way that
$ y_i(\Cal F) \ne q $ for some $\Cal F \in \Cal C$).
The weights $w_{\Cal C}$ are given as products
$\prod_{\Cal F \in \Cal C} w_{\Cal F}$ (see (32))
over frames $\Cal F \in \Cal C$.
The combinatorial coefficients $n_{\Cal C}$ grow
at most as $ C^{\sum_{\Cal F} |\supp \Cal F|}$.
\endproclaim
The proof of (48) is straightforward.
The partition
functions (49)
are then analyzed by cluster expansion method.
See \cite{ZR} for the discussion of the \ps \ technology.
See also \cite{NOZ} showing a simple approach to one
important method
(Cammarota, Dobrushin, Kotecky, Preiss, Seiler,\dots)
of cluster expansions.
Thus, a typical configuration
$x_{\Lambda}$ in the ensemble given by
$Z^q_{\infty}(\Lambda)$ has a small value of $f(x_i)$
(`$x_i$ lives
in the $q$--th well of $U$') for
most $i \in \Lambda$.
One could estimate also the effect of the boundary
condition $q$. Here, it is given in a single point $\infty$
only -- but yet it determines the
prevailing colour $q$ of $\Lambda$
if $q$ is stable ! However the influence,
at $i$,
of the {\bf{position}} of $\infty$
decays exponentially
with $\dist(i,\infty )$.
Our method allows
also an infinite set $Q$ of reference spins $q$.
However, then the `stable' values of $q$ may not exist at all.
More precisely, `finite volume
stability' (observed in a finite torus $\Lambda$
for at least one $q \in Q$)
may disappear in the thermodynamical limit,
for {\bf{any}} $ q \in Q$.
An example of such a situation with an interesting,
`instable'
behaviour in large enough volumes (under any
constant boundary condition $x_i =q$)
is the following model, with $U_1 \equiv 1$
and $ U_2(x_i) = a x_i, a \in \er$:
\remark{Example}
Consider the model
\footnote{For $U_1 \equiv 1$ and $U_2 \equiv 0$ this is
the sine-Gordon
representation of a two dimensional Coulomb gas. See
\cite{BF}, \cite{FS} and the
references given there for more information.}
$$ H(x_{\Lambda}) = \sum_{|i-j| =1}
b (x_i -x_j)^2 \ \ ; \ \
U(x_i) = d \ \cos( x_i U_1(x_i)) + U_2(x_i) \ , \,b > 0 \tag 50
$$
with slowly changing $ U_j$, at low enough temperature.
To guarantee the existence of a stable $q$,
one needs special conditions on
$U_j$. Take $U_1 =1, \,U_2 = 0$. Using our method,
one can show
a coexistence of infinitely many
extremal Gibbs states,
located around the wells of $U$, and an
exponential decay of correlations in these Gibbs states.
I thank to D. Brydges
who explained me the relevance of this example.
\endremark
The aim of this paper was
to reformulate the model
(1) into the language of discrete spin models and to check
the Peierls condition.
The further development and
interpretation of the results thus obtained
surely deserves
an additional study.
\Refs
\widestnumber\key{NOZ}
\ref\key GJ
\by J. Glimm, A. Jaffe
\book Quantum Physics, A Functional Integral
Point of View
\publ Springer Verlag, Berlin
\yr 1981
\endref
\medskip
\ref \key GJF
\by J. Glimm, A. Jaffe, T. Spencer
\paper Phase transition for quantum field
\jour Comm. Math. Phys
\vol 45
\pages 203--216
\yr 1975
\endref
\medskip
\ref\key DS
\by R. Dobrushin, S. Shlosman
\paper Phases corresponding to minima of the
local energy
\jour Select. Math. Sov.
\vol 1
\yr 1981
\pages 317-338
\endref
\medskip
\ref \key I
\by J.Z. Imbrie
\paper Phase diagrams and cluster expansions for $P(\phi)_2$
models.
\jour Comm. Math. Phys.
\vol 82 \pages 261 --304 , 305 --344
\yr 1981
\endref
\medskip
\ref\key M
\by V.A. Malyshev
\paper Cluster expansions in lattice models of statistical
physics
and the quantum theory of fields
\jour Russ. Math. Surveys
\vol 35 \pages 1--62
\yr 1980
\endref
\medskip
\ref\key MM
\by V.A. Malyshev, R.A. Minlos
\book Gibbs Random Fields,
Cluster Expansions
\publ
Kluwer Academic Publishers, Dordrecht, Boston, London
\yr 1991
\endref
\medskip
\ref\key DZ
\by R.L. Dobrushin, M. Zahradn\'{\i}k
\paper Phase diagrams of continuum spin systems
\pages 1--123
\inbook Math. Problems
of Stat\. Phys\. and Dynamics
\publ Reidel
\publaddr
\ed R. L. Dobrushin
\yr 1986
\endref
\medskip
\ref \key PS
\by S.A. Pirogov, Ya.G. Sinai
\paper Phase diagrams of classical lattice systems
\jour Theor. Math. Phys.
\vol 25, 26
\yr 1975, 1976
\pages 1185--1192, 39--49
\endref
\medskip
\ref\key S
\by Ya.G. Sinai
\book Theory of Phase Transitions. Rigorous Results
\yr 1982
\publ Pergamon Press
\endref
\medskip
\ref \key MS
\by R.A. Minlos, Ya.G. Sinai
\paper Phase separation phenomenon at low temperatures
\jour Mat. Sb.
\vol 73
\yr 1967
\pages 375--448
\endref
\medskip
\ref \key MS
\by R.A. Minlos, Ya. G. Sinai
\paper The phase separation phenomenon at low temperature
in some lattice models of a gas
\jour Trudy Mosk. Math. Obsch.
\vol 19
\yr 1968
\pages 113--178
\endref
\medskip
\ref\key Z
\by M. Zahradn\'\i{}k
\paper An alternative version of
Pirogov--Sinai theory
\jour Comm\. Math. Phys\.
\vol 93
\yr 1984
\pages 559--581
\endref
\medskip
\ref\key ZG
\by M. Zahradn\'{\i}k
\paper Low Temperature Gibbs states
and the interfaces between them
\pages 53--74
\inbook Proc.Conf.
Stat. Mech. and Field Theory
\publ Groningen 1985, Lecture Notes Phys. 257
\publaddr
\yr 1986
\endref
\medskip
\ref\key ZR
\by M. Zahradn\'\i k
\paper A short course in the
\ps \ theory
\jour Preprint, 1997
\pages 1--75
\publ
to appear in Rendiconti
di Mathematica, Serie VII,Volume 18
Roma (1998)
\endref
\medskip
\ref\key Ku
\by Ch. Kuelske
\paper Preprint, July 1998, WIAS Berlin
\endref
\medskip
\ref\key NOZ
\by F. Nardi, E. Olivieri, M. Zahradn\'\i k
\paper On Ising model with unisotropic
field
\jour preprint 1998
\endref
\medskip
\ref\key KS
\by R. Koteck\'y, S.Shlosman
\paper First order phase transitions in
large entropy lattice models
\jour Comm. Math. Phys.
\vol 83
\pages 493--515
\yr 1982
\endref
\medskip
\ref \key YP
\by Y.M. Park
\paper Extension of \ps \ theory
of phase transitions to infinite range
interactions
\jour Comm. Math.Phys.
\vol 114
\pages 187-241
\yr 1988
\endref
\medskip
\ref \key FS
\by J. Fr\"ohlich, T. Spencer
\paper The Berezinskii-Kosterlitz-Thouless transition
\inbook Scaling and Selfsimilarity in Physic, Progress in Phys.
\vol 7
\pages 29-139
\publ Birkh\"auser \yr 1983
\endref
\medskip
\ref \key BF
\by D. Brydges, P.Federbush \paper
Debye screening \pages 197-246
\jour Comm. Math. Phys. \vol 73 \yr 1980 \endref
\bye