\centerline{}
\vskip 2cm
\centerline{\bf Mass generation for an interface}
\centerline{\bf in the mean field regime}
\vskip 2.5cm
\centerline{Fran\c cois Dunlop, Jacques Magnen, Vincent Rivasseau}
\vskip .5cm
\centerline{Centre de physique th\'eorique, CNRS, UPR14}
\centerline{Ecole Polytechnique, 91128 Palaiseau Cedex, France}
\vskip 3cm
\centerline{\bf ABSTRACT}
We consider a two dimensional statistical mechanics model of an interface
in three dimensional space with a weak potential tending to
localize the interface near a preferred plane.
For a number of different such potentials we prove that
the two point function decreases exponentially in the mean field
regime where the potential is very flat. We estimate the corresponding
rate of decay.
\vskip 3.5cm
\line {A. 145 01 92 \hfil February 1992}
\vfill\eject
\noindent{\bf I. Introduction}
\medskip
In [DMRR] we bounded the fluctuation of an interface for a gaussian model
with an arbitrarily small attracting potential. In this paper
we study the correlations of this interface and prove
that in the mean field regime corresponding to a very flat potential
well for which the quadratic approximation is valid over a wide range
of values of the interface height, there is an exponential clustering
property and the mass or rate of decay is given by the mean field
value.
This result applies e.g. near a second order wetting
transition in the case of long range forces (these transitions
are usually of first order, but second order transitions
have also been observed [TGVR]). For reviews see [D], [G].
We remark that our results are not optimal
in the sense that when we vary the parameters of the potential
we do not get in this paper exponential decay in the
correct mass in the complete mean field region; we only prove the decay
rigorously in a fraction of what
should be this full mean field region. To improve on this point
is presumably
possible but requires a multiscale analysis together with e.g. the use
of Sobolev inequalities. We postpone this to a future publication. Also
it would be very interesting to investigate regimes in which
non trivial exponents different from the mean field case appear [BHL][KZ].
This requires a rigorous multiscale renormalization group analysis
to compute effective constants. It is presumably not out
of reach of present mathematical techniques [R], but we postpone it
also to future work.
In this paper the main technical tool is, in the language
of field theory, a small field versus large field
expansion which forces to use a non-translation invariant
propagator. This technique is also necessary for several of the most
difficult models in constructive field theory (e.g. [V][CMRV][MRS]), and
the detailed version given in the simpler context of this paper can be
also used as a pedagogical introduction to these more complicated
constructions in field theory.
\medskip
\noindent{\bf II. The gaussian well}
\medskip
Our interface model corresponds to a massless
gaussian measure perturbed by a small interacting potential.
We have to perform rigorously the thermodynamic
limit. Therefore we want to consider first the
massless gaussian measure in a finite volume $\La$, where
for simplicity $\Lambda$ is e.g. a large
square in $\ZZ^{2}$: $\Lambda = \ZZ^{2} \cap [-L,L]^{2} $. Then the
thermodynamic limit is simply $L\to \infty$.
The massless gaussian measure is formally proportional to
$$ {\rm e}^{-(1/2)\sum\limits _{}(h_x-h_y)^{2}}
\prod_{x \in \Lambda } dh_x,\eqno({\rm II.1})
$$
but such an expression is invariant
under global translation of the variables $\{h_{x}, x \in \La\}$. To have a
well defined measure we must break this global invariance, using some
kind of boundary condition at the border of $\La$. A
particularly convenient choice is to use free boundary conditions on
the massive propagator $C $ (with
a value of the mass $m $ which
will be fixed below to precisely
the value expected from the bottom of our gaussian well)
and to make this propagator massless
inside $\La$ by insertion of the suitable ``mass counterterm'';
$${\rm e}^{+(1/2)\sum\limits _{x \in \La}m ^{2} h_x^{2} }.\eqno({\rm II.2})
$$
This rule fits nicely with
the Brydges-Battle-Federbush cluster expansion [B],[R] that we shall use
below. Of course any set of bounded boundary conditions would in fact
lead us to the same thermodynamic limit.
Therefore let us introduce $ C (x,y) $, the ordinary
massive lattice propagator with mass $m ^{2}<< 1$ to be fixed later, which has
the well known Fourier representation:
$$
C (x,y) = {1 \over 2 \pi^{2}}\int_{-\pi}^{\pi } dk_{1}
\int_{-\pi}^{\pi }dk_{2}{ e^{i k\cdot (x-y)}
\over m ^{2} + 2 (2-\cos k_{1} -\cos k_{2}) }
\eqno({\rm II.3})
$$
This propagator has also a
representation as a sum over random paths on the
lattice which in particular proves that it is pointwise positive
in $x$-space. It satisfies the estimate
$$
C (x,y) \le K \cdot \log(1+m^{-1})\cdot e^{- m' \vert x -y
\vert}
\eqno({\rm II.4})
$$
for some positive constant $K$. $m'$ is the optimal decay rate of $C$,
defined by $\cosh m ' = 1+ m^{2}/2$. For small $m$, $m'= m + O(m^{3})$.
Since we are on the lattice, there is in fact anisotropic decay
and one can prove that the worst rate, $m'$, occurs in the lattice
directions (see (A.7)). This point is studied in detail in the Appendix.
In the rest of this paper we use often $K$
as a generic name for such an $m$-independent large constant.
Using free boundary conditions on $C$ we define:
$$ d\mu _{\Lambda } = {1 \over Z _{\La}}
{\rm e}^{+(1/2)\sum\limits _{x \in \La}m ^{2} h_x^{2} } d\mu \quad
;\quad Z _{\La} = \int
{\rm e}^{+(1/2)\sum\limits _{x \in \La} m ^{2} h_x^{2} }d\mu
\eqno({\rm II.5})$$
where $d\mu $ is the normalized measure with propagator $C$ (this
measure can be defined directly in the infinite volume limit).
In the rest of this paper
expectation values such as $<\quad>$ of an observable
always refer to its mean value with
respect to some normalized measure; subscripts
are used to remind the reader of the particular measure
considered. For instance it will be convenient to use the notation $<\ \
> _{\La}$ instead of $<\quad>_{d\mu _{\Lambda }}$.
By an easy gaussian computation the mean value $ $ at any fixed
site $x$ diverges logarithmically as $\Lambda \to \infty$, i.e. as
the thermodynamic limit is performed.
We add now a small interacting potential which tends exponentially
to a constant when $h^{2}$ tends
to infinity but tends to confine $h$ in a neighborhood of 0.
This potential is
$$ V(h) =- \ep [ e^{-{h^{2}\over 2 a^{2}}} -1 ] \eqno({\rm II.6a})
$$
with $a$ and $\epsilon$ both positive (see Fig.1).
\vskip 8cm
\centerline {\bf Fig. 1}
\medskip
We define the normalized measure:
$$d\mu _{V,\Lambda }= {1 \over Z_{V,\La}} d\mu _{\Lambda }
\prod\limits_{x \in \Lambda }e^{-V(h_x)} \eqno({\rm II.6b})
$$
and we will use the notation $<\ \ >_{V,\La}$ for
the expectation value with respect to this measure $d\mu_{V,\Lambda }.$
The regime of parameters which we study is $a >>1$ and $ \ep /a^{2} <<
1$.
The rest of this paper is devoted to a proof that in this regime the two
point function decreases exponentially and to an estimate of the
corresponding mass gap. More precisely we prove:
\medskip
\noindent{\bf Theorem II.1}
\noindent Let $\La = \ZZ \cap [-L,+L]^{2}$, and let
$\{h_{x}\}_{x \in \La}$ be a family of real random variables
distributed according to the probability measure (II.6b), i.e.
the measure
$${1 \over Z_{V,\La}}
{\rm e}^{+(1/2)\sum_{x \in \La}m ^{2} h_x^{2} }
{\rm e}^{-\sum_{x\in \La} V(h_x)} d\mu (\{h_{x}\}_{x \in \La})$$
where $d\mu$ is the gaussian measure of covariance $C(x,y)$ given by
(II.3),
$V(h)$ is given by (II.6) and $m= \sqrt \ep /a$.
Assume $0<\ep \le 1$.
We assume that the potential is such that
$$K\cdot \log (1 +\ep^{-1}) < \sqrt a \eqno({\rm II.7}) $$
where $K$ is a sufficiently
large constant (this means that $a$ is always large
and that if $\ep \to 0$, $a \to \infty$ in a certain way).
Under these conditions the thermodynamic limit
of the correlation functions exists and satisfy an exponential
clustering property (the truncated
correlation functions
decrease exponentially). The decay rate or
effective mass, can be computed
in a systematic expansion
around the decay rate $m'$ of $C$ (which itself by (A.7)
is of order $m= \sqrt \ep /a$ for small $m$).
For instance there exist positive constants $K$ and $c$ such that:
$$
lim_{\La\to \ZZ^{2}}< h_{x}h_{y} >_{V,\La}
\le K \log ( a / \sqrt \ep) e^{-c \sqrt{\ep } \vert x-y\vert/a}
\eqno({\rm II.8})
$$
and $c$ tends to 1 if $\ep$ is fixed and $a \to \infty$.
\medskip
Let us remark first that the technical condition (II.7) under which
the theorem is proved is not expected to be the optimal one under
which we should have exponential clustering within the mean field regime
(i.e. in this model a mass of order $\ep/a^{2}$).
We expect that this theorem in fact holds
under the weaker assumption $K\cdot \log (1 +\ep^{-1}) < a$. The
attentive reader will trace the necessity for a square root in (II.7)
to the third case C) in the proof of Lemma II.5. Here the estimate
that we perform is quite loose. To improve on it we must find a better
upper bound on the $L^{4}$ norm of $h$ in terms of a quadratic norm.
This is provided e.g. by the regular Sobolev inequality
in two dimensions $\Vert h \Vert_{4} \le K \Vert h \Vert_{H_{1}}$
where the $H_{1}$ norm is
$\sqrt {\Vert h\Vert_{2}^{2}+\Vert \nabla h\Vert_{2}^{2}}$. However
we do not use this kind of inequality here since it seems to
require the use of a multiscale analysis, so
that the size of the gradients is adapted to the size of the boxes
in which the inequality is used.
To prove Theorem II.1 we want to analyze the theory with respect to a
lattice ${\bf D}$ which is a regular paving of $\La$ by squares
$\De$ of side $l=a/\sqrt{\ep}$, namely the inverse of the expected mass.
In the squares where the average value of $h$ is less than $\sqrt a$,
which we call the small field region, the quadratic approximation to
the potential which gives a mass $m =\sqrt{\ep } /a$ is valid. In
the rest of this paper the parameter $m $ introduced
in (II.2-3) is therefore fixed to this value $\sqrt{\ep } /a$. In the other
squares, called the large field region, the potential is strictly
above its absolute minimum by a value about $\ep/a$. Taking into account the
number $a^{2}/\ep$ of sites in a square, we remark that large field squares
are rare in probability; they have a suppressing factor $e^{-a}$.
In order to combine these observations into a proof of the theorem,
we are going to perform a cluster expansion with respect to the
lattice ${\bf D}$. Here we go.
For each square we will write
$$
1 = \chi ({1\over \vert \De \vert} \sum_{x \in \De} h_{x}^{4}/a^{2})
+ (1- \chi ({1\over \vert \De \vert} \sum_{x \in \De} h_{x}^{4}/a^{2} ))
$$
$$= \chi (\ep \sum_{x \in \De} h_{x}^{4}/a^{4})
+ (1- \chi (\ep \sum_{x \in \De} h_{x}^{4}/a^{4} ))
\eqno({\rm II.9})
$$
where $\ch$ is a fixed $C ^{\infty}$ function with support in
[0,1], which is one on [0,1/2]. We require also
a rather mild technical condition on $\ch$:
$$
\sup_{h \in \RR } \ {d^{n}\over dh^{n}} \chi (h) \ \le \ K. (n !)^{q}
\eqno({\rm II.10})
$$
for some fixed numbers $K,q>0$ (this is e.g. true with $q=2$ for a
standard shape such as $e^{-1/h}$).
We expand and call $\Ga $ the set of large field squares
$$
1 = \sum_{\Ga \subset \La} \ch_{\Ga}(h) \ ; \ \ch_{\Ga}(h)\equiv
\prod_{\De \not\in \Ga}
\chi (\ep \sum_{x \in \De} h_{x}^{4}/a^{4})
\prod_{\De \in \Ga}(1- \chi (\ep \sum_{x \in \De}
h_{x}^{4}/a^{4} ))
\eqno({\rm II.11})
$$
We insert this expansion (II.11) in the numerator and denominator of the
normalized two point function. Furthermore we develop the potential in
the small field region and combine its quadratic piece
with the counterterm (II.2).
The potential of the theory after this manipulation becomes
$$
V_{\Ga} = -\ep\biggl( \sum_{x\in\Ga} [ e^{-{h_{x}^{2}\over 2a^{2}}} -1 ]+
\sum_{x\not\in\Ga} [ e^{-{h_{x}^{2}\over 2 a^{2}}} -1 +{h_{x}^{2}\over
2 a^{2}} ] \biggr)
$$
$$ = \ep\biggl( \bigl(\sum_{x\in\Ga}\int_{0}^{1}dt
{h_{x}^{2}\over 2 a^{2}} e^{-t\cdot {h_{x}^{2}\over 2a^{2}}} \bigr) -\bigl(
\sum_{x\not\in\Ga} \int_{0}^{1} (1-t) dt
{h_{x}^{4}\over 4 a^{4}}
e^{- t\cdot{ h_{x}^{2}\over 2 a^{2}}} \bigr) \biggr)\eqno({\rm II.12})
$$
(by some slight abuse of notations we will also call $\Ga$ the set of
sites in the squares
of $\Ga$).
The two point function is then given by:
$$
< h_{x}h_{y} >_{V,\La} = { \sum_{\Ga \subset \La}\
\int h_{x}h_{y}\ch_{\Ga}(h) e^{-V_{\Ga}}
{\rm e}^{+(1/2)\sum\limits _{x \in \Ga}m ^{2} h_x^{2} }
d\mu \over \sum_{\Ga ' \subset \La}
\int \ch_{\Ga '}(h) e^{-V_{\Ga '}}
{\rm e}^{+(1/2)\sum\limits _{x \in \Ga '}m ^{2} h_x^{2} }d\mu }
\eqno({\rm II.13})
$$
(remark that in this
formula it is the gaussian measure $Z_{\Ga } d\mu_{\Ga} =
{\rm e}^{+(1/2)\sum\limits _{x \in \Ga}m ^{2} h_x^{2} }
d\mu$ which naturally
appears rather than $d\mu_{\La}$).
We decompose $\Ga$ into connected
components $\Ga_{1}$,..,$\Ga_{n}$ in the following way.
We consider a large factor $M=K \cdot a^{1/4}$,
and we say that two squares of ${\bf D}$ are close if their
minimal distance is smaller than $M/m '$. When two
squares are close in this sense we draw
a link joining them which we call a distance link. Then
a connected component $\Ga_{i}$ is a maximal set of
squares of $\Ga$ connected through such
distance links (hence such that two of them can be linked together
through a chain of squares of $\Ga_{i}$, each of which is close
to the next one in the sense above).
The cluster expansion has to be performed with some care because the
local term ${\rm e}^{+(1/2)\sum\limits _{x \in \Ga}m ^{2} h_x^{2} }
$ cannot be treated as a small perturbation, when we stay
inside a given connected region $\Ga_{i}$ of $\Ga$. However
when we change of connected region, because $M/m'$ is large compared
to the decay length $1/m'$ typical of $C$, we do get a small factor.
We shall perform an expansion which exploits this fact to factorize
the connected components $\Ga_{i}$. We call
$\ch_{S}$ the characteristic function of a set of sites $S$ and
$C_{\Ga}$ the propagator corresponding to the normalized
gaussian measure $d\mu _{\Ga}$.
We want now to compare systematically the covariance $C_{\Ga}$ to $C$ by
means of the resolvent identity:
$$
C_{\Ga}(x,y) = C(x,y)+ \sum_{z}C (x,z)
m ^{2}\ch_{\Ga}(z) C_{\Ga}(z,y) $$
$$= C (x,y)+ \sum_{z}C (x,z) \sum_{i=1}^{n}
m ^{2}\ch_{\Ga_{i}}(z) C_{\Ga}(z,y)
\eqno({\rm II.14})
$$
>From now on let us forget the summation over intermediate points $z$.
We define new objects ${\bf C}^{j,k}$, $j,k=0,1,...n$,
called ``chains'', through the formulas:
$$
{\bf C} ^{0,0} = C +\sum_{p\ge1} \sum_{\scriptstyle
i_{1},...,i_{p} \, \in [1,n]\atop\scriptstyle \ i_{q}\ne
i_{q+1},q=1,...,p-1} C \prod_{q=1}^{p}m ^{2} \ch_{\Ga_{i_{q}}} C
\eqno({\rm II.15})
$$
$$
{\bf C} ^{0,k} = C +\sum_{p\ge1} \sum_{\scriptstyle
i_{1},...,i_{p} \, \in [1,n]\atop\scriptstyle\ i_{q}\ne
i_{q+1},q=1,...,p-1; \ i_{p} \ne k} C
\prod_{q=1}^{p}m ^{2} \ch_{\Ga_{i_{q}}} C
\eqno({\rm II.16})
$$
$$
{\bf C} ^{j,0} = C +\sum_{p\ge1} \sum_{\scriptstyle
i_{1},...,i_{p} \, \in [1,n]\atop\scriptstyle \ i_{q}\ne
i_{q+1},q=1,...,p-1; \ i_{1}\ne j} C
\prod_{q=1}^{p}m ^{2} \ch_{\Ga_{i_{q}}} C
\eqno({\rm II.17})
$$
$$
{\bf C} ^{j,k} = C +\sum_{p\ge1} \sum_{\scriptstyle
i_{1},...,i_{p} \, \in [1,n] \atop\scriptstyle \ i_{q}\ne
i_{q+1},q=1,...,p-1; \ i_{1}\ne j, i_{p} \ne k} C
\prod_{q=1}^{p}m ^{2} \ch_{\Ga_{i_{q}}} C \quad {\rm if\ } 0\ne
j\ne k \ne 0
\eqno({\rm II.18})
$$$$
{\bf C} ^{j,j} = \sum_{p\ge1} \sum_{\scriptstyle
i_{1},...,i_{p} \, \in [1,n] \atop\scriptstyle \ i_{q}\ne
i_{q+1},q=1,...,p-1; \ i_{1}\ne j, i_{p} \ne j} C
\prod_{q=1}^{p}m ^{2} \ch_{\Ga_{i_{q}}} C \quad {\rm if\ } j\ne 0.
\eqno({\rm II.19})
$$
We apply the identity (II.14) repeatedly and
we obtain, in the sense of operators:
$$
C_{\Ga} = {\bf C} ^{0,0} + \sum_{p\ge1}
\sum_{j_{1},...,j_{p} \, \in [1,n]}
\ {\bf C} ^{0,j_{1}} \prod_{k=1}^{p}
\biggl( (m ^{2}\ch_{\Ga_{j_{k}}}C_{\Ga_{j_{k}}}m ^{2}
\ch_{\Ga_{j_{k}}})\
\ {\bf C} ^{j_{k},j_{k+1}}\biggr)
\eqno({\rm II.20})
$$
with the convention that $j_{p+1}\equiv 0$. When $\Ga$ is made of a
single connected component $\Ga_{1}$
these formulas simplify a lot; we have $C_{\Ga} =C_{\Ga_{1}} $ and
$$
{\bf C} ^{0,0} = C +
C m ^{2} \ch_{\Ga_{1}} C
\eqno({\rm II.21})
$$
$$
{\bf C} ^{0,1} = {\bf C} ^{1,0} = C \ ;\ {\bf C} ^{1,1}=0
\eqno({\rm II.22})
$$
$$
C_{\Ga} = {\bf C} ^{0,0} +{\bf C} ^{0,1}
(m ^{2}\ch_{\Ga_{1}}C_{\Ga_{1}}m ^{2}\ch_{\Ga_{1}})\ {\bf C} ^{1,0}
\eqno({\rm II.23})
$$
Then we define $h_{0} $ and $h_{i}$, $i=1,..,n$ as
independent gaussian random variables with respective covariance
${\bf C} ^{0,0}$ and $C_{\Ga_{i}}$.
The corresponding normalized gaussian measures are called respectively
$d\mu_{0}(h_{0})$ and $d\mu_{i}(h_{i}) \equiv
d\mu_{\Ga_{i}}(h_{i})$, $i=1,...,n$.
If we perform the substitution
$$h(x) = h _{0}(x) + \sum_{i=1}^{n}\, \sum_{y}\
{\bf C} ^{0,i}(x,y) m^{2}\ch_{\Ga_{i}}(y) h_{i}(y)
\eqno({\rm II.24})
$$
we have
$$
d\mu_{\Ga}(h) =
{1 \over P_{\Ga}} \prod_{i=0}^{n} d\mu_{i}(h_{i}) e^{Chains }\ ; \
P_{\Ga}= \int \prod_{i=0}^{n} d\mu_{i}(h_{i}) e^{Chains }
\eqno({\rm II.25})
$$
where
$$ Chains \equiv (1/2)
\sum\limits_{1\le i \le n, 1\le j \le n}
m^{2}\ch_{\Ga_{i}}h_{i}\ ({\bf C} ^{i,j}) m^{2}\ch_{\Ga_{j}}h_{j} \
. \eqno({\rm II.26})$$
(II.25) means that the integral of any function $f$ of the variable
$h=\{h_{x}, x\in \La\}$ with respect to
the measure $d\mu_{\Ga}$ is equal to the same function integrated with
respect to the right hand side measure (II.25) if the substitution
(II.24) is made in $f$ (we used the fact that ${\bf C} ^{0,i}
(x,y) = \ {\bf C} ^{i,0} (y,x)$).
Since what appears in (II.13) is the measure
$$
{\rm e}^{+(1/2)\sum\limits _{x \in \Ga}m ^{2} h_x^{2} }
d\mu = Z_{\Ga} d\mu_{\Ga} \eqno({\rm II.27})
$$
we have to compute the normalizing ratio $Z_{\Ga} /P_{\Ga}$. This
is done using the following Lemma:
\medskip
\noindent{\bf Lemma II.1}
\noindent We have
$$
Z_{\Ga} /P_{\Ga} = \prod _{i=1}^{n} Z_{\Ga_{i}}
\eqno({\rm II.28})
$$
\medskip
\noindent{\bf Proof } We rewrite $P_{\Ga}$ as
$$ P_{\Ga}= \int e^{ Chains } d\mu_{0} \prod_{i=1}^{n} d\mu_{\Ga_{i}}(h_{i})
$$
$$ =\int e^{Chains }\prod_{i=1}^{n}\biggl( Z_{\Ga_{i}}^{-1}\ e^{(1/2)
\sum\limits_{x\in \Ga_{i}}
m^{2}h_{i}^{2} (x)} d\mu(h_{i}) \biggr)
\eqno({\rm II.29})
$$
using the fact that the factor $Chains$
does not depend on $h_{0}$ and (II.27). It remains therefore to prove that
$$ \int e^{(1/2)
\sum\limits_{1\le i \le n, 1\le j \le n}
m^{2}\ch_{\Ga_{i}}h_{i} \ ({\bf C} ^{i,j})m^{2}\ch_{\Ga_{j}} h_{j} }
\prod_{i=1}^{n}\biggl( e^{(1/2)
\sum\limits_{x\in \Ga_{i}}
m^{2}h_{i}^{2}(x) } d\mu(h_{i})\biggr)$$
$$= \int e^{(1/2) \sum\limits_{x\in \Ga} m^{2}h^{2}(x)} d\mu = Z_{\Ga}.
\eqno({\rm II.30})
$$
This is just an exercise in expanding both sides of (II.30) into power
series in $m^{2}$, integrating by Wick's theorem and identifying the
cycles of propagators on both sides. Indeed on both
sides of (II.30) we get cycles made of insertions $m^{2}\ch_{\Ga}$
joined by propagators $C$, but on the left hand side
of (II.30) these cycles are simply decomposed according to whether or not
successive insertions of $\Ga$ are of the type $\Ga_{i}$-$\Ga_{i}$ or
$\Ga_{i}$-$\Ga_{j}$ with $j\ne i$. Identity (II.30) is therefore
quite the analogue for cycles of the resolvent identity (II.20).
Using this lemma, we can rewrite (II.13) as
$$
< h_{x}h_{y} >_{V,\La} = { \sum_{\Ga \subset \La}\ \prod_{i=1}^{n}
Z_{\Ga_{i}} \int h_{x}h_{y}\ch_{\Ga}(h) e^{-V_{\Ga}(h)}
\prod_{i=0}^{n} d\mu_{i}(h_{i}) e^{Chains }
\over \sum_{\Ga ' \subset \La} \prod_{i '=1}^{n'}
Z_{\Ga'_{i '}}
\int \ch_{\Ga '}(h) e^{-V_{\Ga '}}
\prod_{i '=0}^{n'} d\mu_{i '}(h_{i '}) e^{Chains' }}
\eqno({\rm II.31})
$$
where
$$ Chains' \equiv (1/2)
\sum\limits_{1\le i ' \le n ', 1\le j ' \le n '}
m^{2}\ch_{\Ga '_{i '}}h_{i '}\ ({\bf C} ^{i ',j '}) m^{2}
\ch_{\Ga '_{j'}}h_{j '} .\eqno({\rm II.32})$$
is the analogue of (II.26) for the decomposition of $\Ga '$
as the union of connected components $\Ga ' _{i '} $, $i ' = 1,...,n
'$.
The reason for all this rewriting is that the various propagators
${\bf C} ^{j,k}$ $j,k=0,...n$
as defined by (II.15-19) are well defined through
absolutely convergent series and have good decay properties. More
precisely:
\medskip
\noindent{\bf Lemma II.2}
\noindent
For any $\ze '>0$ (arbitrarily small) there exists $K>0$ (depending
on $\ze '$ but not on $m$) such that
$${\bf C} ^{j,k} (x,y) \le K \log (1+ m^{-1})
e^{-m ' (1-\ze ') \vert x-y \vert}\ \quad
\forall j \in [0,n], \forall k \in [0,n]
\eqno({\rm II.33})
$$
\medskip
\noindent{\bf Proof } We use the fact that $dist (\Ga_{i},
\Ga_{j})\ge M/ m' $ together
with the estimate (II.2). The conditions that two consecutive indices in
(II.15-19) have to be different ensure that for $M$ large enough we
can extract a small factor for each of these terms, and keep an
exponential decrease $e^{-m' (1-\ze ') \vert u-v \vert}$ between the
ends $u$ and $v$ of each $C $ piece. The triangular inequality and the
convergence of geometric series with ratio smaller than one completes
the proof, if we ensure that $ \log (1+ m^{-1}) e^{- M (1-\ze
')} << 1 $, a condition which is satisfied by our choice
$M =K\cdot a^{1/4}$ and the condition (II.7), which ensures
$ \log (1+ m^{-1}) < a^{1/2} $ at large $a$.
\medskip
We shall also need a lemma to control the normalization factors
$Z_{\Ga_{i}}^{-1}$ which appear in (II.31):
\medskip
\noindent{\bf Lemma II.3}
\noindent We have, for some constant $c$ independent of $m $
$$
Z_{\Ga}^{-1} \ \le \ e^{c\cdot N(
\Ga ) \log(1+m ^{-1}) }
\eqno({\rm II.34})
$$
where $N(\Ga) =m ^{2} \vert
\Ga\vert$ is the number of of squares in $\Ga$,
if $\vert \Ga\vert$ is the number of sites in $\Ga$.
\medskip
\noindent {\bf Proof} We have, using the explicit formula for the
normalization of a gaussian integral:
$$
Z_{\Ga}^{-1} =\biggl( \det
(1 - m ^{2}\ch_{\Ga}C \ch_{\Ga}) \biggr)^{1/2} =
e^{(1/2) \sum_{n \ge 1} {1 \over n} Tr
\bigl(m ^{2}\ch_{\Ga}C \ch_{\Ga}\bigr)^{n} }
$$
$$ \le
e^{(1/2) \vert \Ga \vert \sum_{n \ge 1} {1 \over n} \sup_{x\in \Ga}
\bigl(m ^{2}C \bigr)^{n}(x,x) }$$
$$ =
e^{(1/2) \vert \Ga \vert \sum_{n \ge 1} {1 \over n}
{1 \over 2 \pi^{2}}\int_{-\pi}^{\pi } dk_{1}
\int_{-\pi}^{\pi }dk_{2}\bigl( { m ^{2}
\over m ^{2} + 2 (2-\cos k_{1} -\cos k_{2}) }\bigr)^{n} }$$
$$ \le
e^{cm ^{2} \vert \Ga \vert \log (1+m ^{-1}) } \eqno({\rm II.35})
$$
In the first inequality we have used the fact that the propagator
(II.3) is pointwise positive to increase the sums, which were
restricted to $\Ga$ to the full volume $\RR^{2}$ (except for one,
which fixes translation invariance and gives the volume factor
$\vert\Ga\vert$). Then we used Fourier analysis. The last inequality
is easy; the term with $n=1$ gives explicitly the logarithmic factor,
and the other terms are uniformly bounded by $m^{2}\sum_{n\ge 2}O(1)/n^{2} $.
It remains to perform a cluster expansion on (II.31). This is done
both on the numerator and denominator of (II.31). We use the
formalism of Brydges-Battle-Federbush (see [B],[R] for reviews). We will
first describe this cluster expansion and give an outline of the main
important points to understand its structure and the reasons for
its convergence. Then we state the main result in the form of Lemma II.4 below
for which we give a more detailed proof of convergence.
We consider e.g. the numerator of (II.31). First
we list in an arbitrary order the set $U$ made
of all squares of the small field region plus the $n$ elements
$\Ga_{1},...,\Ga_{n}$. Then we
introduce a parameter $s$ which in every propagator $C $ in any
of the ${\bf C} ^{j,k}$ terms of (II.15-19)
decouples the first element of $U$ from the rest. For instance if this
first element is called $\De$ we write:
$$
C (s)= \ch_{\De} C \ch_{\De} +
(1-\ch_{\De})C (1-\ch_{\De}) $$
$$+ s\bigl( \ch_{\De}C (1-\ch_{\De}) +
(1-\ch_{\De}) C \ch_{\De}\bigr)
\eqno({\rm II.36})
$$
When we insert this interpolated covariance into (II.31), the measure
$d\mu_{0}$, the factor $e^{Chains}$ and the definition of $h$ through
(II.24) in the factor $h_{x}h_{y}\ch_{\Ga}(h) e^{-V_{\Ga}(h)}$ change.
Remark that the measures $d\mu_{i}$ or the normalization factor
$\prod Z_{\Ga_{i}}$ do not change; indeed these factors are already
factorized over the connected large filed regions.
We have to check that inserting $C(s)$ instead of $C$ in
the definition (II.15) of ${\bf C} ^{0,0}$ gives still a measure of
positive type, so that we have a well defined functional integral.
This is obvious.
Then we expand the numerator of (II.31) at first order around $s=0$
using the Taylor formula with integral remainder. The term at $s=0$
decouples $\De$ from the rest; the
remainder term couples $\De$ to some other element of
$U$, $\De '$ (which again can be a small field square or some $\Ga_{i}$)
by means of some explicit propagator $C $ inside the definition of some
chain ${\bf C} ^{j,k}$. Then we consider $\De \cup \De '$ (or
$\De \cup \Ga_{i}$) as a single new entity in $U$ and iterate. This
cluster expansion is described in detail in [B],[R]. Rather than to repeat all
the details here, we will simply insist on all the differences
with the standard case treated in [B],[R] of a gaussian measure
with propagator $C $ perturbed by a polynomial interaction
(such as $h^{4}$). The differences are
the non polynomial nature of the ``interaction'' factor
$\ch_{\Ga}(h) e^{-V_{\Ga}(h)}$ and the particular
structure of the chains ${\bf C} ^{j,k}$ which do not reduce to a
single propagator $C$.
Let us address these differences now. Each derivation $d/ds$ which couples
some element $\De \in U$ to $\De '\in U$
creates an explicit propagator $C $ in some chain with its
two ends in the prescribed objects $\De$ and $\De '$. This is
called a cluster link between $\De$ and $\De '$. In the standard
case [R], using integration by parts, we have at the two ends of the
propagator a functional derivation $\de \over \de h$
acting either on the sources or on the exponential of
the interaction, which creates a so called ``derived''
vertex (such as $4h^{3}$ in the case of an $h^{4}$ Ginzburg-Landau
model) localized at
this end. In our case let us also call a derived vertex the result
of a functional derivative $\de \over \de h$ applied to the
interaction term which in our case is $\ch_{\Ga}(h) e^{-V_{\Ga}(h)}$.
Then a cluster link appears as
a propagator $C $ created by a $d/ds$
derivation, with its ends in $\De$ and $\De '$, which
lies in some chain ${\bf C} ^{j,k}$. We have to decsribe
the factors which lie at the end of this chain. An end of chain
corresponding to an index $j$ in $ [1,n]$ has simply a factor
$m^{2}\ch_{\Ga_{j}}h_{j}$ hooked to it (this is true both for
the two ends of the chains in the
exponential term $e^{Chains}$, or for one end of the chains
of the type ${\bf C} ^{0,k}$ or ${\bf
C} ^{j,0}$, $j,k \in [1,n]$ which appear in the replacement of
$h$ in (II.28) by formula (II.24)). An end of chain corresponding
to a 0 index (such as both ends of a ${\bf C} ^{0,0}$ chain
or one end of a ${\bf C} ^{0,k}$ or ${\bf
C} ^{j,0}$, $j,k \in [1,n]$ chain) instead has a ``derived vertex''
${\de \over \de h } \ch_{\Ga}(h) e^{-V_{\Ga}(h)}$
hooked to it (after the functional derivative
has been computed we have to replace $h$ by its value (II.24) to
reexpress this derived vertex in terms of the fields $h_{0}$ and
$h_{i}$, $i=1,...n$).
Indeed this is directly the result in the case of the end with 0 index
of a ${\bf C} ^{0,k}$ or ${\bf
C} ^{j,0}$, $j,k \in [1,n]$ chain and in the case of the two ends
of a ${\bf C} ^{0,0}$ chain this is the result of the functional
derivative $\de \over \de h $ hooked to the end, after
an integration by parts on $h_{0} $. Indeed from (II.24) we see that
the action of a functional derivative $\de \over \de h $
on a function of $h$ is the same
as the action of $\de \over \de h_{0}$ on this function, reexpressed
in terms of $h_{0}$ and $h_{i}$, $i=1,...n$.
This description of the cluster link is correct up to two further
remarks. In a small number of cases (at most two for a two point
function) the derivation at the end of a chain instead of producing
a derived vertex may hit the source fields $h_{x}$ or $h_{y}$. Also
a derived vertex may be hit by further derivations, hence in the most
general case what lies at the end of a chain is really a multiply derived
vertex common to several links, or a source. This is
completely standard, although remark
that in the case of a polynomial interaction such as $h^{4}$,
a derived vertex can be rederived
only at most a fixed number of times. In our case the intercation
factor is non-polynomial and a derived vertex can be hit
an arbitrary number of times. However remark
that these derivations must be associated to farther and farther squares
and the longer and longer distance factors in (II.33)
will control the associated combinatoric problem.
A slight difference with the standard case which may worry the reader
is that a chain ${\bf C} ^{j,k}$ is made of a sum over $n$ of terms
containing an arbitrarily large number $n$ of propagators $C$.
How to control the combinatoric of choosing on which of these $n$
terms a $d/ds$ derivation may act? This is easy. Because of the
inductive rules of the Battle-Brydges-Federbush cluster expansion,
only one propagator $C $ in a given chain sandwiched between two
characteristic functions of two given large field region $\Ga_{i}$
and $\Ga_{j}$ can be explicitly derived (indeed later these two
regions are treated as a single block). Therefore paying a factor 2
per such sandwiched propagator we may decide whether it will be
derived or not. This sloppy bound is then easy to control because
each such sandwiched propagator gives an arbitrarily small factor
(see the proof of Lemma II.2).
Finally there is a subtlety here that we have to take into
account which is caused by our definition of distance links between
large field cubes. Because of this definition, a large field region
has a halo of radius $M /m'$ where other large field cubes cannot
enter. This is an analogue of the hardcore condition which is familiar
in cluster expansions.
In the end of the cluster expansion we have sets of squares connected
together through explicit cluster links, and the connected large field regions
$\Ga_{i}$ which are connected together by distance links.
Taking all connections into account (both
cluster links and distance links) we have a collection of
connected sets of squares $E_{1}$, ...,$E_{r}$ called
clusters or polymers (whose
union must be all of $\La$). We can discard the trivial clusters
made of empty small field squares, whose amplitude is 1, and
consider the non-trivial ones, whose union is no longer $\La$.
We claim now
that our functional integral is factorized over these sets. Indeed
the only reason for non-factorization may come now from the
normalizing factor in (II.28): but this factor is
$\prod_{i=1}^{n}Z_{\Ga_{i}}^{-1}$, and since
each $\Ga_{i}$ is contained entirely
in a single $E_{k}$, we conclude that the $E_{k}$ are factorized.
Applying this process to the numerator
and the denominator of (II.31) we obtain
$$
< h_{x}h_{y} >_{V,\La} = {\sum_{\{ E_{j} \}, E_{j}{\rm\ generalized
\ disjoint \ family} }
A_{x,y}(E_{1})\prod_{j \ge 2 } A_{\emptyset}(E_{j}) \over
\sum_{\{ E '_{k} \}, E '_{k} {\rm \ generalized \ disjoint \ family}}
\prod_{k } A_{\emptyset}(E '_{k})}
\eqno({\rm II.37})
$$
where $A_{x,y}(E_{1})$ is the amplitude of the connected cluster $E_{1}$
containing the sources $x$ and $y$ (by parity they have to lie in the same
cluster if the interaction is even), and
$A_{\emptyset }(E_{j})$ or $A_{\emptyset }(E_{k})$ is a vacuum amplitude
associated to the non-trivial cluster $E_{j}$, $j\ge 2$ or $E_{k}$.
The condition $E '_{k} {\rm \ generalized \ disjoint \ family}$ means that the
$E_{j}$ are disjoint in the ordinary sense $plus$ the fact that any
large field square in $E_{j} $ is separated by at least $M /m'$
from any large field square in $E_{j'} $, $j\ne j'$.
More generally one can derive
a formula which generalizes (II.37) to the computation of any
correlation function $S$ of $N$ external sources
$h_{i_{1}},...,h_{i_{N}}$,
as a sum of products of amplitudes, where in the numerator the union of
all amplitudes has to contain all external sources.
To complete the result of the cluster expansion it is standard
that we need only to prove that non-trivial clusters are small so that
they can be resummed. Let us introduce a measure $N(E)$
of the size of the cluster $E$ which is equal to the number of small
field squares plus $M^{2}$ times the number of large field squares in $E$.
Then we will prove below:
\medskip
\noindent{\bf Lemma II.4}
\noindent
Each vacuum amplitude is bounded so that (if $O$ is an arbitrary origin)
$$
\sum_{E / O\in E} A_{\emptyset} (E) e^{N( E ) } \le 1
\eqno({\rm II.38})
$$
Furthermore the amplitude containing the two external sources at
sites $x$ and $y$ has exponential
decay so that
$$
\sum_{E / x,y \in E} A_{x,y} (E) e^{ N( E ) } \le K e
^{-(1-\ze '')m' \vert x-y \vert}
\eqno({\rm II.39})
$$
where $\ze ''$ tends to 0 if $a \to \infty$.
\medskip
We could establish more general ``tree decay'' between sources for
amplitudes containing more than two sources. They are not necessary
however for the proof of our main theorem. In fact we will limit
ourselves to study vacuum amplitudes and prove as usual
a result stronger then (II.38), namely one in which the precise constant
$e$ can be replaced by any other fixed constant, if $a$ is large
enough and $\ep / a^{2}$ is small enough. The result (II.39) follows
easily by evaluating $A_{x,y} (E)$ in the same manner than a vacuum
amplitude, but taking into account that $x$ and $y$ have to be
connected through chains of cluster or distance links. The cluster
links give directly exponential decay of the form (II.39) and
each distance link give a small factor which tends to 0 as $a \to \infty$
(Lemma II.6 below). Combining both effects we get (II.39).
Assuming (II.38-39) the proof of the theorem is achieved
by a standard Mayer expansion on (II.37) which removes the
constraints of generalized disjointness
in (II.37) and accordingly computes the
normalized functions. This is completely standard (see e.g. [B],[R]).
The only subtlety has to do with the nature
of the generalized disjointness condition. Because of the halo in this
condition
one gets an exponential of the number of squares in the
region $E$ plus the surrounding halo in the process of resummation of
clusters linked to $E$ through Mayer links (or overlapping
conditions). This factor is then compensated thanks to our definition of
$N(E)$, so that Lemma II.4 does indeed ensure the convergence
of the Mayer expansion.
If the amplitudes in (II.22) have
exponential tree decay between the sources, this exponential tree
decay follows for the truncated correlation functions computed by the Mayer
expansion. In particular in the case of the two point function
this argument achieves the proof of our main theorem.
We concentrate therefore on the summation of vacuum amplitude
which contains the origin, and give a detailed proof of (II.38).
The cluster expansion has produced a certain number of
explicit fields $h_{i}$, $i=0,1,...n$, hooked
to the ends of $C^{j,k}$ chains to which the derived cluster links
$C$ belong, or produced by functional derivatives
$\de/\de h$ acting on $\ch_{\Ga} e^{-V_{\Ga}}$. Let us explicit
the structure of all these terms.
The outcome of the Brydges-Battle-Federbush expansion is indexed by
a tree $T$ [2,3] of cluster links between nodes called
$N_{1}$,..., $N_{n+1}$, which can be the large field
regions $\Ga_{1},...,\Ga_{r}$
or the small field squares
$\De_{1},...,\De_{r'}$ which together form the support of $E$. We have
$n+1=r+r'$. The
tree $T$ is therefore made of $n=r+r'-1$ links $L_{l}$,
$l=1,...,r+r'-1$, each of which contains a $C$ propagator
with ends in two different nodes $S_{j_{l}}$ and $S_{k_{l}}$.
(By the Brydges-Battle-Federbush
process one has $n\ge 1$ except for trivial clusters whose values is 1).
This propagator is in fact part of a chain $C_{l}$ which together with
$C$ may contain other propagators and has in addition some attached
factors at the end. Let us describe this in more detail.
The $l$-th chain $C_{l}$ has its ends in squares $\De_{e_{l}}$
and $\De_{f_{l}}$ which are in $E$ but can be of course different from
$S_{j_{l}}$ and $S_{k_{l}}$. If the chain is of type $C^{j,k}$ with
$j,k >0$ we have fields $h_{j}$ and $h_{k}$ attached to these ends
and this is the end of the story. The set of all fields $h_{i}$, $i>0$,
attached to such chains form a monomial which we call $S$.
But if the chain is of type
$C^{0,0}$, $C^{0,j}$ or $C^{k,0}$ we have instead of explicit
fields $h_{0}$ functional derivatives ${\de \over \de h(x)}$
$ \ch_{\Ga} e^{-V_{\Ga}}$ attached to the ends
with 0 index, which have to be applied to the factor
$ \ch_{\Ga} e^{-V_{\Ga}}$.
We have to perform these functional derivatives, and
in the corresponding fields produced, we have to replace $h$ by
formula (II.24), which again may or may not produce eventually chains
which link $\De_{e_{l}}$
and $\De_{f_{l}}$ to a set of
final squares $\De^{s}_{g_{l}}$ and $\De^{s}_{h_{l}}$
that contain the final fields $h_{j}$, $j>0$
associated to the chains $C^{j,0}$ which occur in (II.24).
>From the explicit form (II.9) and (II.12) of $\ch_{\Ga} e^{-V_{\Ga}}$
we conclude below that at most five fields can be produced by a
functional derivative, hence the index $s$ takes at most five values
(see Fig.2).
\vskip 8cm
\centerline {\bf Fig. 2}
\medskip
If we put together all the functional derivatives $\de \over \de h
(x)$ which act in a given square $\De$ they must be of the form
$\bigl(\sum_{x\in \De}{\de \over \de h (x)}\bigr)^{n_{\De}}$, with
$\sum_{\De \in E} n_{\De} \le 2n-2$ (because there are at most two ends
per chain $C_{l}$, hence at most $2n-2$ such derivatives).
We compute the action of these derivatives and obtain:
$$\prod_{\De}\bigl(\sum_{x\in \De}{\de \over \de h
(x)}\bigr)^{n_{\De}} \ch_{\Ga} e^{-V_{\Ga}} =
\prod_{\De }
\sum_{\scriptstyle n_{\De,1}, n_{\De,2}\atop\scriptstyle
n_{\De,1}+n_{\De,2}=n_{\De}} { n_{\De} \choose n_{\De,1} }
$$
$$\sum_{p_{\De,1}\le n_{\De,1}} Q_{p_{\De,1}}(\{h(x)\}, x\in \De)
\ps ^{(p_{\De,1})}
({1\over \vert \De \vert} \sum_{x \in \De} h_{x}^{4}/a^{2})
R_{n_{\De,2}}(\{h(x)\}, x\in \De) e^{-V_{\Ga}}.\eqno({\rm II.40})
$$
where $\ps = \ch $ if $\De \not\in \Ga$ and $\ps = 1-\ch $ if $\De \in \Ga$.
$Q$ is a polynomial of order $4p_{\De,1}-n_{\De,1}$ and $R$ is a
polynomial of order at most $5 n_{\De, 2}$ times an exponential
of a negative quadratic form in the variables $(\{h(x)\}, x\in \De)$.
This result is obtained using the form (II.12) of $V_{\Ga}$.
The exact formulas for $Q$ and $R$ are tedious to write down, because
the Leibniz formulas to derive products after many iterations become
complicated. However to bound the outcome of (II.41) we need only
to keep track of the general structure of $Q$ and $R$. First we
bound by 1 the exponential
of the negative quadratic form in the variables $(\{h(x)\}, x\in \De)$
remaining in $R$, and we bound
all the factors $t$ or $1-t$ and the integrals
$\int_{0}^{1}dt$ coming from (II.12) by 1. Second we use the condition
(II.10) to bound $\prod_{\De }{ n_{\De} \choose n_{\De,1} } \ps ^{(p_{\De,1})}
({1\over \vert \De \vert} \sum_{x \in \De} h_{x}^{4}/a^{2}) $
by $K^{n} \prod_{\De} (n_{\De}~!)^{q} \Omega_{\Ga}(h)$, where we use
that $\sum_{\De} n_{\De} \le 2n-2$, and
we define $\Omega_{\Ga}(h)=\prod_{\De \not\in \Ga}
\om_{1} ({1\over \vert \De \vert} \sum_{x \in \De} h_{x}^{4}/a^{2})
\prod_{\De \in \Ga}(\om _{2} ({1\over \vert \De \vert} \sum_{x \in \De}
h_{x}^{4}/a^{2} )) $, where $\om_{1}$ is the characteristic function
of [0,1] and $\om _{2}$ the characteristic function of [1/2, $+\infty$].
Finally in $Q$ and $R$ we replace $h$ by its value (II.24). This is
the step which generates the squares $\De^{s}_{g_{l}}$ and
$\De^{s}_{h_{l}}$;
we see that as announced there is only at most 5 such fields per
functional derivative.
In this way up to a numerical factor
$$ K^{n} \prod_{\De} (n_{\De}~!)^{q} \eqno({\rm II.41})
$$
and up to the explicit value of the chains
$C^{j,k}$, the functional integral that remains to be bounded
has the form of a certain function $F= P\cdot\Omega_{\Ga} e^{-V_{\Ga}}$
of the fields, where $P= \prod_{i=0}^{n}
\prod_{j=1}^{N(i)} h_{i}(x_{j})$ is the explicit product of all
the fields produced in all the polynomials $Q$, $R$, and in the first
monomial $S$ directly attached to the
chains in (II.31) and considered above.
We want now to show that to each cluster link $L_{l}$ is associated
a small factor. This factor will come either
from the spatial decay of the links in Lemma II.2 or to
some small factors attached to the fields produced by functional
derivatives
when they act in the small field region. Let us explain this
point in detail, considering again
the particular form (II.11-12) of $\ch_{\Ga} e^{-V_{\Ga}}$.
The outcome of a functional derivative ${\de \over \de h(x)}$
in a square $\De_{e_{l}}$
or $\De_{f_{l}}$ depends on whether this square is a large or small
field square. In the case of a small field square
every derivation on $ \ch_{\Ga} e^{-V_{\Ga}}$ acts either on $\ep(h^{4}/a^{4})$
or on $\ep(h^{4}/a^{4})e^{-h^{2}/a^{2}}$ and produces
either $\ep(h^{3}/a^{4})$ or $\ep(h^{3}/a^{4})e^{-h^{2}/a^{2}}$ or
$\ep(h^{5}/a^{6})e^{-h^{2}/a^{2}}$. In (II.40) we said that we
bound the negative
exponentials by one and keep only the produced fields. Therefore the
outcome of one functional derivation is a set of at most 5 fields,
which have then to be expanded according to (II.24). If the square is a
large field square, i.e. belongs to $\Ga$, the functional derivative
acts either on $\ep(h^{4}/a^{4})$ or on $\ep(h^{2}/a^{2})e^{-h^{2}/a^{2}}$
and produces either $\ep(h^{3}/a^{4})$ or
$\ep(h/a^{2})e^{-h^{2}/a^{2}}$. Again recall that the negative
exponential is bounded by one.
First we remark that from the form of these vertices
to each independent summation over $x$ in a square $\De$ we can
associate a factor $\ep/a^{2} = 1/\vert \De \vert$ coming either from
the factors $\ep$ and $a^{-2}$ or $a^{-4}$ in (II.12) or from
(II.11). In this sense every such summation is properly normalized.
Then in addition to these factors we have $in$ $the$ $case$ $of$ $a$
$vertex$ $produced$ $in$ $a$ $small$ $field$ $square$ an additional
factor $a^{-2}$ for an $h^{3}$ vertex and $a^{-4}$ for an $h^{5}$
vertex. In order to take correctly
into account these additional factors let us
introduce a new notion.
We say that a cluster link is a small field link if $S_{j_{l}}$ and
$S_{k_{l}}$ are both in the small field region. In the other case we
call it a large field link. For every large field link, by our rule
we must have ${\rm dist } \{ S_{j_{l}}, S_{k_{l}} \} \ge M/m'$. Therefore
we can extract a factor at least $e^{-M/5}= e^{-Ka^{1/4}/5}$
from every such link, still keeping four fifths of the
spatial exponential decay in the $l$
chain for other purposes.
In the case of a small field link both $S_{j_{l}}$ and
$S_{k_{l}}$ have to be at a distance at least $M/m'$ from any
large field square of $\Ga$. We call the link a regular link
if both squares $\De_{e_{l}}$
and $\De_{f_{l}}$ are in the small field region and all the
fields produced by formula (II.24) are of the type $h_{0}$.
In the converse case we call the link irregular. For an irregular link
at least one of the squares $\De_{e_{l}}$, $\De_{f_{l}}$,
$\De^{s}_{g_{l}}$ or $\De^{s}_{h_{l}}$ has to belong to $\Ga$.
Therefore using the triangular inequality plus one fifth of the
spatial decrease of the $C^{j,k}$ chains we can
again extract a factor at least $e^{-M/5}= e^{-Ka^{1/4}/5}$ for such
an irregular link.
The regular links have all their produced fields of the $h_{0}$
type, hence these fields belong to the production square
$\De_{e_{l}}$ or $\De_{f_{l}}$. For these regular links we can
use the additional powers of $a^{-1} $ described above to attribute a small
factor $1/\sqrt a$ to each end of
the link and a small
factor $1/\sqrt a$ to each of the produced fields.
For irregular links or large field links we have for large $a$
$e^{-Ka^{1/4}/5} << a^{-16}$. Therefore we can attribute
also a small
factor $1/\sqrt a$ to each of the produced fields, and a factor $1/a$
to the link $l$ and to each of the $C^{0,j}$ links produced by formula
(II.24) (there are at most five of them per end of $l$ hence at most
ten of them). Finally we use condition (II.7) which ensures
that $\log(1+m^{-1})< \sqrt a $. Each link $l$ and each of the links
$C^{0,j}$ produced has therefore in this way
an associated factor $\sqrt a /a =
1/\sqrt a$, instead of the factor $\log(1+m^{-1})$ of Lemma II.2, and
each vertex produced is normalized
by $\ep /a^{2}$ and each
field produced is normalized by a factor $1/\sqrt a$.
It remains to extract, using (II.4) and Lemma II.2
the remaining exponential decay from the explicit propagators.
We call $d_{l}$ the distance factor which is the sum
of the minimum distance between $S_{j_{l}}$ and $S_{k_{l}}$, plus
if necessary,
the minimum distance between $S_{j_{l}}$ and $\De_{e_{l}}$, $S_{k_{l}}$
and $\De_{f_{l}}$, and between $\De_{e_{l}}$
and each $\De^{s}_{g_{l}}$, $\De_{f_{l}}$
and each $\De^{s}_{h_{l}}$. We can extract, using (II.4) and Lemma II.2
the remaining four fifths of the
corresponding exponential decay from the explicit propagators
associated to the $l$-th chain in Fig. 2.
In this way we can bound
the explicit cluster links by a factor
$$ (K/\sqrt a)^{n} \prod_{l=1}^{n} e^{-(1-\ze ')(4 m'/5) \cdot
d_{l}} \eqno({\rm II.42})
$$
where $K$ is again some constant independent of $m$, and
each vertex produced is normalized
by $\ep /a^{2}$ and each field produced is normalized by a factor $1/\sqrt a$.
The exponential decrease in (II.40)
will be used later both to sum over the various locations
of the squares and regions which form $E$ and also to control
``local factorials'' generated by integration of the
fields produced by the functional derivatives, and by the combinatoric
of Leibniz's formula for derivations of products.
The sums over all combinatoric factors associated with the
various choices in the Leibniz formula are certainly bounded again by a factor
$K^{n} \prod_{\De} (n_{\De}!)^{q}$ for $K$ and $q$ large enough.
This simplifies our bound; taking (II.41) and (II.42) into account
we collect a multiplicative factor
$$ (K/\sqrt a )^{n}
\prod_{\De} (n_{\De}!)^{2q}
\prod_{l=1}^{n} e^{-(1-\ze ')(4 m '/5) \cdot
d_{l}} \eqno({\rm II.43})
$$
(with some enlarged value for $K$)
and we have still to bound the supremum
over functional integrals
of functions $F'= P'\cdot\Omega_{\Ga} e^{-V_{\Ga}}$ in which $P'$ is now
a monomial (without sums and prefactors)
$\prod_{x} (h_{i}(x)/\sqrt a)^{p_{i}(x)} $.
The functional integration over $F'$ is bounded using a Schwartz
inequality to separate the fields in $P'$
from the rest. This means that we write
$$
\int F'(\{h(x)\}) \prod_{i=0}^{n} d\mu_{i} $$
$$\le
\biggl(\int \prod_{x} (h(x)/\sqrt a) ^{2p(x)}
\prod_{i=0}^{n} d\mu_{i} \biggr)^{1/2}
\biggl( \int \Omega_{\Ga}e^{-2V_{\Ga}} \prod_{i=0}^{n} d\mu_{i} \biggr)^{1/2}
\eqno({\rm II.44})
$$
(since $\Omega_{\Ga}=\Omega_{\Ga}^{2}$).
We bound first the second functional integral in (II.44).
\medskip
\noindent{\bf Lemma II.5}
\noindent
The second functional integral in (II.41) satisfies the bound:
$$
\int \Omega_{\Ga}e^{-2V_{\Ga}} \prod_{i=0}^{n} d\mu_{i} \le
K^{n} \prod_{\De \in E \cap\Ga}
e^{- \sqrt a \cdot \log a}
\eqno({\rm II.45})
$$
where $K$ is a positive constant.
\medskip
\noindent{\bf Proof }
We remark first that the positive hence
potentially dangerous piece of $e^{-2V_{\Ga}}$
corresponding to the small field squares of $E$ (see (II.12))
can be exactly bounded,
using the small field condition $\om_{1}$ in
$\Omega_{\Ga}$ by $e^{2 r'}$,
where we recall that $r'$ is the number of small field
squares in $E$. This factor can be absorbed in the constant $K$ in (II.45).
Let $\De$ be a square of the large field region $\Ga$, and $K$
be some large constant. Either
A)
$$
\sup_{x \in \De } \vert h_{x}\vert \le 2K \sqrt{a \log a }
\eqno({\rm II.46})
$$
or
B)
There is a site $x \in \De$ such that
$$
\vert h_{x} \vert \ge 2K \sqrt{a \log a }
\eqno({\rm II.47})
$$
and
$$\inf_{x \in \De } \vert h_{x}\vert \ge K \sqrt{a \log a }
\eqno({\rm II.48})
$$
or
C)
There is a site $x \in \De$ and a site $y\in \De$ such that
$$
\vert h_{x} \vert \ge 2K \sqrt{a \log a }
\eqno({\rm II.49})
$$
$$
\vert h_{y} \vert \le K \sqrt{a \log a }
\eqno({\rm II.50})
$$
In the first case we write, using the large field condition $\om_{2}$:
$$
1/2 \le { 1\over \vert\De\vert } \sum_{x\in\De} h_{x}^{4}/a^{2}
\le \biggl({ 1\over\vert \De\vert }
\sum_{x\in\De} h_{x}^{2}/a^{2}\biggr)
(\sup_{x \in \De }\vert h_{x}\vert )^{2}\eqno({\rm II.51})
$$
Therefore
$$ \biggl( \sum_{x\in\De} h_{x}^{2}/a^{2}\biggr) \ge
{\vert \De \vert \over 8 K ^{2} a (\log a )} = {a \over 8 \ep K ^{2}
(\log a )}
\eqno({\rm II.52})
$$
The potential then gives the small factor. Indeed we can assume
$K^{2}(\log a )/a \le 1$, since $a$ is large; then $\vert h_{i}\vert
\le 1$ for $x \in \De$ and since $e^{-t}-1 \le -t/e$ for $t \le 1$,
we conclude that
$$
e^{-V_ {\De}} = e^{\ep \sum_{x \in \De} e^{-h_{x}^{2} /a^{2}} -1 }
\le e^{-\ep \sum_{x \in \De} h_{x}^{2} /e.a^{2} }
\le e ^{- {a \over 8e K ^{2} (\log a )}} \le e^{- \sqrt a \cdot \log a}
\eqno({\rm II.53})
$$
if $a$ is big enough.
In the case B) we use the fact that
$\inf_{x \in \De } \vert h_{x}\vert \ge K \sqrt{a \log a }
$
to obtain directly that
$$
e^{-V_ {\De}} = e^{\ep \sum_{x \in \De} e^{-h_{x}^{2} /a^{2}} -1 }
\le e^{-\ep \sum_{x \in \De} (1/e)\inf \{ h_{x}^{2} /a^{2} , 1 \} }
$$
$$\le e ^{- (a/e) \inf \{ K ^{2} (\log a ), a \} }
\le e^{- \sqrt a \cdot \log a}
\eqno({\rm II.54})
$$
In the case C) we use the fact that
the gaussian measure gives a small factor when two sites
not too far apart have very different values. More precisely we write
$$
\int d\mu \sum_{x} \sum_{y}\ch ( \vert h_{x} \vert \ge 2K \sqrt{a \log a })
\ch ( \vert h_{y}\vert \le K \sqrt{a \log a })
$$
$$\le (a^{2}/\ep)^{2} e ^{- 2 \sqrt a \cdot\log a}\le
e ^{- \sqrt a \cdot \log a}\eqno({\rm II.55})
$$
Indeed the gaussian distribution corresponding to two sites
$x$ and $y$ after integrating on the others behaves as
$e^{-c (h_{x}-h_{y})^{2}/2 \log \vert x-y \vert}$, where $c$
is some constant, and $\log \vert x-y \vert \le \log a/\sqrt{\ep} \le
\sqrt a $, if $a$ is large, using the
condition (II.7) in the Theorem. If we take $c K^{2} >2 $ (by
increasing $K$) the first inequality in (II.55) follows. The second
inequality is again obtained using (II.7) since $a^{4}/\ep^{2}
< a^{4}e^{(2/K)\sqrt a}$ is beaten by the small factor
$e ^{- \sqrt a \cdot \log a}$.
In every case the proof of (II.45)
is achieved.
Let us return to the first functional integral in (II.44).
We perform this gaussian integration explicitly.
Remark that all
fields $h_{i}$ produced for $i=1,...,n$ are in fact of the type
$m^{2}\ch_{\Ga_{i}}(x)h_{i}(x)$, that is they are localized inside
$\Ga_{i}$ and they are multiplied by a factor $m^{2}$. The result of Wick's
theorem is a certain number of graphs with propagators $C^{0,0}$ or
$C_{\Ga_{i}}$. We use first a Schwartz inequality again to bound
$C^{0,0}(x,y)$ or $C_{\Ga_{i}}(x,y)$ respectively by
$(C^{0,0}(x,x))^{1/2}(C^{0,0}(y,y))^{1/2}$ or
$(C_{\Ga_{i}}(x,x))^{1/2}(C_{\Ga_{i}}(y,y))^{1/2}$.
Then we use the following bound
\medskip
\noindent{\bf Lemma II.6}
$$
C^{0,0} (x,x)
\le K \cdot \log(1+m^{-1})
\eqno({\rm II.56})
$$
$$
C_{\Ga_{i}} (x,x)
\le K \cdot \sup \{ \log(1+m^{-1}) , \log (1+ d(x, \partial \Ga_{i}) \}
\eqno({\rm II.57})
$$
\medskip
\noindent{\bf Proof } (II.56) follows from (II.33). For (II.57)
using the random path expansion of the propagator it is easy to bound
$C_{\Ga_{i}} (x,x)$ by $C_{\RR^{2}-\{y\} } (x,x)$, where $y$ is the
point closest to $x$ in the complement of $\Ga$.
We can consider again that the distribution for the two sites
in a massless gaussian measure after integration of the others is
$e^{-c (h_{x}-h_{y})^{2}/2 \log \vert x-y \vert}$. The distribution
for $C_{\RR^{2}-\{y\} } (x,x)$ is massless except for a factor
$e^{-(1/2) m^{2}h_{y}^{2}}$. Using this factor and integrating on $y$
we obtain the distribution for $h_{x}$ and achieve the proof of (II.57).
Then we remark that in the product (II.44) each field to integrate has
an associated normalizing factor $1/\sqrt a$. Using the fact that at
large a $\log (1 + m^{-1}) < \sqrt a << a$ we can use
these normalizing factors to compensate again all
the factors $\log (1 + m^{-1})$ produced by gaussian integration and
Lemma II.6.
Using an other fifth of the exponential decrease
$\prod_{l=1}^{n} e^{-(1-\ze ') (4m '/5)\cdot d_{l}}$ we can beat the product
of all factors $\log (1+ d(x, \partial \Ga_{i}))$ generated by
(II.57). Indeed each field $h_{i}(x)$ is at the end of a chain
whose last propagator has length at least equal to $d(x, \partial \Ga_{i})$.
It remains to bound the local factorials generated by Wick's theorem
in the gaussian integration of the fields.
Our final fields to contract are localized in squares
of the type $\De^{s}_{g_{l}}$ or $\De^{s}_{h_{l}}$, and a naive bound would
involve factorials of the number of fields localized
in such squares. But using an other fifth of the remaining
exponential decrease $\prod_{l=1}^{n} e^{-(1-\ze ') (3m '/5) \cdot
d_{l}}$ it is easier to choose, in the Wick's contraction process,
the squares $\De _{e_{l}}$ or $\De _{f_{l}}$ which contain the initial vertex
from which the chain to the contracted field emanate.
In this way the factorials of Wick's contractions
is bounded by
$K^{n}\prod_{\De} (n_{\De}!)^{5}$ (since there are at most five fields
produced per vertex).
Using a standard volume argument we know that from a remaining
fifth of the exponential
decay $\prod_{l=1}^{n} e^{-(1-\ze ')(2 m '/5) \cdot
d_{l}}$ we can extract a factor
$K^{n_{\De}}\prod_{\De} (n_{\De}!)^{-q'}$ where $q'$ can be made as
large as we want [3, Lemma III.1.3]. This is because $n_{\De}$ can
become large only when we have more and more distant squares
$\De '$ hooked to $\De$ by the cluster expansion. In particular
we can take $q' > 2q +5$. In this way
we can compensate
the local factorials $\prod_{\De} (n_{\De}!)^{2q+5}$.
Finally the sums over the positions of the squares in $E$
is also made using the last fifth of the exponential decay
$\prod_{l=1}^{n} e^{-(1-\ze ') (m'/5) \cdot
d_{l}}$; the summation is made according to the natural
tree structure $T$ which is the outcome of the
Battle-Brydges-Federbush process. We have also to sum over the regions
$\Ga_{i}$, $i=1,..,r$, knowing one of their squares. This is done
using the distance links, and there is therefore an associated factor
$ \prod_{\De \in E \cap\Ga}
M^{2}$. This factor is compensated by the one of Lemma II.5, since
$M = K \cdot a^{1/4}$.
In order to have the small factor in $N(E)$ in (II.38) we must also
extract a factor $e^{-M^{2}}$ from each small field square in $E$.
This can be extracted from a fraction of
the factor $\prod_{\De \in E \cap\Ga}
e^{- \sqrt a \cdot \log a}$ in (II.45), since
$e^{M^{2}} =e^{K^{2}\sqrt a } << e^{+ \sqrt a \cdot \log a} $.
After having extracted this factor
we get a final geometric sum over the number $n$ of elements
in $E$ of a term certainly bounded by
$$ K^{n} a^{-n/2}
\eqno({\rm II.58})
$$
where $K$ is independent of $a$. Taking $a$ large enough we get a
geometric series with ratio as small as we want.
This proves in particular the condition (II.38), hence achieves the proof
of Lemma II.4 and of the convergence of our cluster expansion.
\vfill\eject
\medskip
\noindent{\bf III. The wetting potential with a wall}
\medskip
In order to model the presence of the wall in a wetting problem
we consider now different potentials which are not symmetric, but for
which the method of the last section applies with minor modifications.
We shall not detail as much the proof in these asymmetric cases.
\medskip
\noindent{\bf A) The linear exponentials}
\medskip
The first example that can be considered
is a potential made of two competing linear exponentials (Fig.3):
$$
V (h) = \ep^{2}/2 + \ep e^{-\al h} - (1/2) e^{-2\al h}
\eqno({\rm III.1})
$$
\vskip 8cm
\centerline {\bf Fig. 3}
\medskip
The ``wall''
is the region $h<0$, which is exponentially suppressed. On the side
$h>0$ we have fast asymptoticity of the potential to a constant, just
as in the preceding model (but the asymptotic value is reach in a
linear instead of quadratic exponential way).
We find that the minimum is at $h^{*}$ such that $e^{-\al h^{*}}=\ep$.
If we write $\hat h = h - h^{*}$, the analogue of the Taylor
formula (II.12) is
$$
e^{-V(h)} = \exp \{- {\ep^{2} \over 2} [e^{-\al \hat h}-1]^{2} \}
\eqno({\rm III.2a})$$
$$=\exp \{ - {\ep^{2}\al^{2}\hat h^{2} \over 2} +
\ep^{2}\al^{3}\hat h^{3} \int_{0}^{1}(-{e^{-\al t \hat h}\over 6} + {2
e^{-2\al t \hat h}\over 3}) {(1-t)^{2}\over 2} dt \} \ .
\eqno({\rm III.2b})
$$
The first form will be used in the large field region out of the well,
the second form is suited for a cluster expansion in the small
field region (inside the well).
Therefore the mass, in the regime where the gaussian well is
very flat, is expected to be $m = \ep \al$. For a fixed value
of $\ep$, we can make the mass very small by letting $\al \to 0$. In
this sense the parameter $\al$ plays the r\^ole of $a^{-2} $ in the
previous section.
Here we need a slightly more complicated small field condition
which states both that the field is approximately inside the well, and
that the exponential in the potential is under control. For instance
we can bound the Taylor remainder
in (III.2b) using a Schwartz inequality:
$$\ep^{2}\al^{3}\hat h^{3} \int_{0}^{1}(-{e^{-\al t \hat h}\over 6} + {2
e^{-2\al t \hat h}\over 3}) {(1-t)^{2}\over 2} dt \}$$
$$ \le
\biggl({1\over \vert\De\vert}\int_{\De} \al^{2}\hat h^{6} \biggr)^{1/2}\ .
\biggl({1\over \vert\De\vert}\int_{\De} \sup \{1, e^{-4\al h}\}\biggr)^{1/2}
\eqno({\rm III.3})
$$
Therefore we can choose as the small field condition
for a square $\De$ of side $m^{-1}$:
$$
\ch ({1\over \vert\De\vert}
\int_{\De} \al^{2}\hat h^{6} )
\ch ({1\over \vert\De\vert} \int_{\De}\sup \{1, e^{-4\al h}\} )\ .
\eqno({\rm III.4})
$$
so that in the small field region we control the Taylor remainder.
We need again as technical condition analogue of (II.7)
a bound which is not optimal:
$$
K \log (1+\ep^{-1}) < \al^{-1/3}
\eqno({\rm III.5})
$$
Remark again
that for technical reasons we cannot cover with our a single scale
analysis the complete mean field region, which we expect here to be
given by a condition of the type
$$K \log (1+\ep^{-1}) < \al^{-2/3} \ . \eqno({\rm III.6})
$$
Indeed the vertices produced in the small field region are ${\de \over \de h}
\ep^{2}\al^{3}\hat h^{3}$ times exponentials which are controlled
by the second part of the
small field condition. Using some simple
power counting, such a vertex
is evaluated by $\al \log^{3/2} (1+\ep^{-1}) $ after integration in a
square $\De$. In order to correspond to a small factor, condition (III.6)
is enough and (III.5) is more than sufficient. However in the large
field region we need to gain a factor small enough to compensate the
normalization. The worst case for the large
field region is when the first function $\ch$ in (III.4) is replaced
by $(1-\ch)$, the other case giving a much smaller factor. But
performing an analysis similar to that of Lemma II.5, in case C)
we obtain a small
factor in $e^{- \al^{-2/3}/\log (1+\ep^{-1})}$, which has
to beat a normalization factor similar to (II.35), hence condition
(III.5) is necessary.
We can state:
\medskip
\noindent{\bf Theorem III.1}
\medskip
Theorem II.1 holds true if $V(h)$ in (II.6a)
is taken as in (III.1) with condition (III.5)
and $m=\ep\al$.
\medskip
\noindent{\bf B) The compact wall}
\medskip
A slight modification of the exponential wall (III.1) is to add a
cutoff function $\prod_{x} \et (h_{x})$ where $\et$ is a $C^{\infty}$
function which is 0 for $h_{x}<0$ and 1 if $h_{x}\ge 1$, in order
to model better the fact that the interface cannot penetrate the wall
(see Fig.4).
\vskip 8cm
\centerline {\bf Fig. 4}
\medskip
The rules of the expansion are unchanged and the only additional
technical problem is when a functional derivative hits an $\et$
function.
This produces vertices of the type $\et' (h) = \hat h + h^{*}$. Such
fields correspond to the large field region, and do not change the
range of validity of the expansion. Therefore Theorem III.1 also
holds for this model.
\medskip
\noindent{\bf C) The Van-der-Waals potential}
\medskip
The Van der Waals potential for molecular attraction ([D],[G]) leads to the
consideration of wetting potentials of the type:
$$
V= -{\ep \over 2h^{2}} + {1\over 3h^{3}} +{\ep^{3}\over 6}
\quad {\rm if} \ \ h\ge 0
$$
$$V = +\infty \quad {\rm if} \ \ h < 0 \eqno({\rm III.7})
$$
The minimum of the potential is at $h^{*}=\ep^{-1}$. The expected mass
is $m=\ep^{5/2}$ and goes to 0 with $\ep$. There is here a single
parameter. We can put again $\hat h = h-h^{*}$ and write (for $h\ge 0$):
$$
V= + {\ep^{5}\over 2} \hat h^{2}
+\int_{0}^{1}({12 \ep \over t^{5}(\hat h + \ep^{-1})^{5}} - {20
\over t^{6}(\hat h + \ep^{-1})^{6}}) {(1-t)^{2}\over 2} dt \} \ .
\eqno({\rm III.8})
$$
We can write again large and small field conditions using (III.8).
However the situation is simpler here since there
is only one parameter. Our cluster expansion therefore does
apply simply for $\ep$ small enough:
\medskip
\noindent{\bf Theorem III.2}
\medskip
Theorem II.1 holds true if $V(h)$ in (II.6a)
is taken as in (III.7) with $\ep$ small enough
and $m=\ep^{5/2}$.
\medskip
The case of a Lennard-Jones potential is exactly similar, but with
different values of the exponents in (III.7).
\medskip
\noindent{\bf D) More general potentials }
\medskip
>From the discussion of the specific examples above it is clear
that our method generalizes to any sufficiently smooth
potential with a single absolute minimum strictly below all
othe local minima, in the regime where the corresponding well is
very flat, i.e. the gaussian approximation is a good approximation for
a rather large range of values of the interface height. The exact
limits of validity of the cluster expansion depends of the shape
and parametrization of the curve giving the potential, so that we
do not state a precise general theorem. Smoothness of the potential
everywhere is presumably
not physically essential but for our method it is a useful technical
ingredient, even in the ``large field region'', because it allows analytic
computation of the functional derivatives, which in our cluster
expansion can act in this large field region. To treat the case of
non-smooth potential is presumably possible but certainly requires
some additional technical work.
\vfill\eject
\noindent{\bf Appendix: Estimate of the lattice covariance}
\medskip
\noindent The inverse covariance of mass $m$ on $\ZZ^{2}$ is defined
as
$$
C^{-1} (x,y) = (4+m^{2} )\de_{x,y} - \de_{\vert x-y\vert,1}
\eqno({\rm A.1})$$
so that
$$
{1 \over 2} \sum_{x,y \in \ZZ^{2}} \ph (x) C^{-1}(x,y) \ph(y)
= {1 \over 2} \sum_{\vert x-y \vert =1} \bigl(\ph(x) -\ph (y)
\bigr)^{2}+ {m^{2}\over 2} \sum_{x \in \ZZ^{2}} \ph (x) ^{2}
\eqno({\rm A.2 })$$
where the sum is over $pairs$ of nearest neighbors and
$$
<\ph (x) \ph (y) > = C(x,y) = {1 \over 4\pi^{2} } \int_{-\pi}^{\pi} dk_{2}
\int_{-\pi}^{\pi} dk_{1} {e^{ik_{1}(x_{1}-y_{1})+ik_{2}(x_{2}-y_{2})}
\over m^{2} +4 -2 \cos k_{1} -2 \cos k_{2}}
\eqno({\rm A.3 })$$
\medskip
\noindent{\bf Lemma}
\noindent Let $m \le 1$ and $x_{1}\ge x_{2}\ge 0$. Let $m_{1}$ and
$m_{2}$ be the functions of $m,x_{1},x_{2}$ defined through
$$
m_{1}x_{1}+m_{2}x_{2}= \sup _{\cosh m'_{1}+ \cosh m'_{2} = 2+ m^{2}/2}
\ \{ m'_{1}x_{1}+m'_{2}x_{2}\}
\eqno({\rm A.4})
$$
Then
$$
C\bigl((x_{1},x_{2}),(0,0)\bigr) = O\bigl(\log
(m(1+x_{1}))^{-1}\bigr)
\quad \ {\rm if}\ mx_{1}\le 1
$$$$
C\bigl((x_{1},x_{2}),(0,0)\bigr) \ = \ O(1) {e^{-m_{1}x_{1}-m_{2}x_{2}}
\over \sqrt{mx_{1}}}
\ \quad \ {\rm if}\ mx_{1}\ge 1
\eqno({\rm A.5})
$$
which implies $\forall x,y$
$$
C(x,y) \le c (\log m^{-1} )\cdot e^{-m' \vert x-y\vert}
\eqno({\rm A.6})
$$
where $\vert x-y\vert $ is the Euclidean distance and $m'$ is defined
by
$$
\cosh m' = 1+ m^{2}/2
\eqno({\rm A.7})
$$
\medskip
\noindent{\bf Remark } The slowest decay, at $\pi /4$, is given by $e^{-m''
\vert x-y\vert}$ with$$
\cosh {m'' \over \sqrt 2} = 1+ m^{2}/4
\eqno({\rm A.8})
$$
\medskip
\noindent {\bf Proof } $x_{1}=0$ is easy. We suppose $x_{1}\ge 1$ and
begin by shifting the contour of integration in $k_{2}$. For any
$m_{2}