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BODY
\documentstyle [12pt]{article}
\title{Nontrivial Fixed Points and Screening in the Hierarchical
Two-Dimensional Coulomb Gas}
\author{G. Benfatto\\ Dipartimento di Matematica, II Universit\`a di Roma\\
00133 Roma, Italy
\and
J. Renn\thanks{{Associate of the Department of Physics, Harvard University.
Supported by Gruppo Nazionale per la Fisica Matematica, CNR, Italy.}}
\\Department of Physics, Boston University\\Boston, MA 02215, USA
}
\date{}
\begin{document}
%\bibliographystyle{unsrt}
\maketitle
\begin{abstract}
We show the existence and asymptotic stability of two fixed points of the
renormalization group transformation for the hierarchical two-dimensional
Coulomb gas in the sine-Gordon representation and temperatures slightly
greater than the critical one. We prove also that the correlations at the
fixed points decay as in the hierarchical massive scalar free field theory,
that is as $d_{xy}^{-4}$. We argue that this is
the natural definition of screening in the hierarchical approximation.
\end{abstract}
%
%\input macros
%
%%%%% FONTS

\font\msxtw=msxm10 scaled\magstep1
\font\euftw=eufm10 scaled\magstep1
\font\msysm=msym10
\font\msytw=msym10 scaled\magstep1
\font\amit=cmmi7 \def\sf{\textfont1=\amit}

%%%%% GREEK

\let\a=\alpha \let\b=\beta \let\d=\delta  \let\e=\varepsilon
\let\f=\varphi \let\g=\gamma \let\iu=\upsilon
\let\k=\kappa    \let\l=\lambda \let\m=\mu    \let\n=\nu
\let\o=\omega \let\p=\pi \let\r=\rho \let\s=\sigma \let\t=\tau
\let\th=\theta \let\z=\zeta
\let\D=\Delta \let\F=\Phi \let\G=\Gamma \let\L=\Lambda \let\O=\Omega
\let\P=\Pi \let\Ps=\Psi \let\Si=\Sigma \let\Th=\Theta

%%%%% VECTORS

\def\V#1{\vec#1} \def\kk{{\V k}} \def\oo{{\V\o}} \def\rr{{\V r}}
\def\uu{{\V\iu}} \def\vv{{\V v}} \def\xx{{\V x}}  \def\yy{{\V y}}
\def\zz{{\V z}}
\def\OO{{\V\O}}

%%%%% CALLIGRAPHIC AND BOLD

\def\E{{\cal E}} \def\ET{{\cal E}^T} \def\LL{{\cal L}}
\def\SS{{\cal S}} \def\VV{{\cal V}}

\def\bq{{\bf q}} \def\bh{{\bf h}}
\def\bT{{\bf T}} \def\bL{{\bf L}} \def\bK{{\bf K}}

%%%%% SYMBOLS

\def\RR{\hbox{\msytw R}} \def\RRs{\hbox{\msysm R}}
\def\CC{\hbox{\msytw C}} \def\NN{\hbox{\msytw N}}
\def\pb{\bar\psi}
\def\pt{\tilde\psi} \def\bt{\tilde b} \def\vt{\tilde v} \def\dt{\tilde d}
\let\pd=\partial \def\Pd{\V\pd}

\def\ie{\hbox{\it i.e.\ }}
\def\tende#1{\vtop{\ialign{##\crcr\rightarrowfill\crcr
              \noalign{\kern-1pt\nointerlineskip}
              \hskip3.pt${\scriptstyle #1}$\hskip3.pt\crcr}}}
\def\gtopl{\hbox{\msxtw \char63}}
\def\ltopg{\hbox{\msxtw \char55}}
\def\Halmos{\hfill \vrule height10pt width4pt depth2pt \vskip.5cm}
\def\virg{\quad,\quad} \def\mezzo{\frac{1}{2}}
\def\bsl{$\backslash$}
\def\comb(#1,#2){\left(\begin{array}{c}#1\\#2\end{array}\right)}

%%%%% ABBREVIATIONS

\def\be{\begin{equation}}\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}\def\eea{\end{eqnarray}}
\def\bean{\begin{eqnarray*}}\def\eean{\end{eqnarray*}}
\def\bfr{\begin{flushright}}\def\efr{\end{flushright}}
\def\bc{\begin{center}}\def\ec{\end{center}}
\def\ba#1{\begin{array}{#1}} \def\ea{\end{array}}
\def\bd{\begin{description}}\def\ed{\end{description}}
\def\bv{\begin{verbatim}}
\def\nn{\nonumber} \let\lb=\label
\def\dsty{\displaystyle}

%%%%% OTHER DEFINITIONS

\def\pref#1{(\ref{#1})}
\def\Dim{{\bf Proof -\ \ }}
\def\msy#1{\hbox{\msytw #1}}
\def\got#1{\hbox{\euftw #1}}
\def\asp{\\ [5pt]} \def\lasp{\\ [10pt]}

\newtheorem{lemma}{Lemma}[section]
\newtheorem{theorem}{Theorem}[section]
\newtheorem{prop}{Proposition}[section]
\renewcommand{\theequation}{\arabic{section}.\arabic{equation}}
%
%\input coul_1
%
\setcounter{section}{0}\setcounter{equation}{0}
\section{Introduction} \lb{s1}
In the last ten years the renormalization group ideas have been extensively
applied to the two dimensional Coulomb gas of identical charges $\pm e$, in
order to rigorously understand the so called Kosterlitz-Thouless phase
transition \cite{KT}.

For inverse temperatures $\b$ larger than the critical one, $\b_c$, and
small activity $\l$, it has been proved that there is no screening \cite{FS,%
MKF}. This result is strictly related to the fact that, in the field
theoretical representation of the model (the so called sine-Gordon
representation), the effective potential goes to zero as the scale goes to
infinity \cite{DH}.
All these results are valid also in the hierarchical approximation of the
model (see Sect. \ref{s2}), where they are obtained much easier \cite{BGN,MF}.

For $\b<\b_c$ screening is generally expected to be found, but this property
has been
proved in the exact model  only for $\b$ very small \cite{BF}. However, in the
case of the hierarchical approximation, a weak form of screening has been
proved for $\b\hbox{\msxtw .}\b_c$ \cite{MF}.

In this paper we study the hierarchical Coulomb gas in the sine-Gordon
representation in the region $\b\hbox{\msxtw .}\b_c$, by analyzing
the renormalization group transformation $\bT$ of the effective potential.
We prove that $\bT$ has two nontrivial asymptotic stable fixed points, which
have the following screening property: the two-charges truncated correlations
decay as $d_{xy}^{-4}$ as the (hierarchical) distance $d_{xy}$ goes to
infinity, which is the decaying behaviour of the truncated correlations
of the massive hierarchical scalar field. Our analysis also suggests that the
effective potential for the hierarchical Coulomb gas on scale $1$ is in the
domain
of attraction of one of the fixed points, if the activity is small enough, so
that in this case as well screening should be observed.

The existence of the nontrivial fixed points was proved in \cite{MF} with a
different technique, which does not use the sine-Gordon representation. The
proof of \cite{MF} extends to more general models, but allows one to study
only
the simple (not truncated) correlations. This is why the analysis of \cite{MF}
was restricted to the screening for fractional charges; in fact, for the
fractional charges, the truncated correlations coincide with the simple ones.

Finally, we want to stress that the technique used in this paper is
essentially based on the bound discussed in the Appendix, which is a bound for
the Ursell coefficients of a system of arbitrary charges sitting in the same
point and interacting with a potential $cQ_i Q_j$. We were unable to find
this estimate, which we think is interesting by itself, in the literature.
%
%\input coul_2
%
\setcounter{section}{1}\setcounter{equation}{0}
\section{The hierarchical model} \lb{s2}
Let $Q_j, j\in \NN$, be a sequence of compatible pavements of $\RR^2$ made of
squares of side size $\g^j$, where $\g\ge 2$ is an integer. To each $\D\in
Q_j$ we associate a gaussian variable $z_\D$ such that
\be \E(z_\D^2)= \frac{1}{2\p} \log \g \virg \E(z_\D z_{\D'})=0\;{\rm if}\;
\D\not=\D' \lb{2.1}\ee
Then we define, $\forall x\in \RR^2$:
\be \f_x = \sum_{k=0}^\infty z_{\D_x^{(k)}} \lb{2.2}\ee
where $\D_x^{(k)}$ is the tessera of side size $\g^k$ containing $x$.

Given $x,y\in \RR^2$, let $h_{xy}$ be the smallest integer such that there
exists a $\D\in Q_{h_{xy}}$ containing both $x$ and $y$. We shall call $d_{xy}
\equiv \g^{h_{xy}}$, the side size of $\D$, the hierarchical distance between
$x$ and $y$. By using \pref{2.1} it is easy to see that:
\be \E((\f_x-\f_y)^2) = \E
\left( \sum_{k=0}^{h_{xy}-1}[z_{\D_x^{(k)}}-z_{\D_y^{(k)}}]^2 \right) =
\frac{1}{\p} \log d_{xy} \lb{2.3}\ee
which justifies the claim that $\f_x$ is a reasonable approximation of the
two-dimensional zero mass scalar field. In the following we shall denote
the corresponding Gaussian measure by $P(d\f)$.

If $v(z), z\in \RR^1$, is a real function such that
\be v(0)=0 \virg v(z)=v(-z) \lb{2.4}\ee
and $\L$ is a finite volume belonging to $Q_R$, for some $R>0$, we shall
consider the measure
\bea \m_v^\L (d\f) &=& \frac{1}{Z_v^\L} P(d\f) \prod_{\D\in Q_0\cap\L}
e^{v(\f_\D)} \lb{2.5}\\
Z_v^\L &=& \int P(d\f) \prod_{\D\in Q_0\cap\L} e^{v(\f_\D)} \eea
where $\f_\D$ is the constant value of the field on the tessera $\D$.

The choice
\be v(\f) = \l (\cos(\a\f)-1) \lb{2.6}\ee
corresponds to the hierarchical Coulomb gas in the volume $\L$ with activity
$\l/2$, charges $\pm e$ and temperature $\b^{-1}$, such that
\be \b e^2 = \a^2 \lb{2.7}\ee
For more details on this point see \cite{BGN}, where a rescaled field was used,
instead of \pref{2.2}.

Another interesting choice is
\be v(\f)=-m^2\f^2 \equiv u_m(\f) \ee
which should give rise to the two dimensional hierarchical scalar field of
mass $m$. It is important to remark, however, that this is not a good
approximation of the massive scalar field. In fact it is easy to show that
\be \lim_{\L\to\RRs^2} \int \m_{u_m}^\L(d\f) \f_x\f_y \propto d_{xy}^{-4}
\lb{2.10}\ee
in disagreement with the exponential decay of the massive scalar field (a
similar property is valid in other hierarchical models, see \cite{GaK},
chapter 4, exercise 2).

This observation will be very relevant in the following, since it implies that
the hierarchical Coulomb gas should have power decaying correlations also in
a screened phase, but with a power equal to $4$ independently of $\b$.

Let us now define the renormalization group transformation.

If $F(z)$ is a function on $\RR$ and $<\cdot>_v^\L$ denotes the expectation
w.r.t. the measure \pref{2.5}, then
\be <F(\f_0)>_v \equiv \lim_{\L\to\RRs^2} <F(\f_0)>_v^\L =
<\bL_{\bT^{k-1}v}\cdots \bL_v F(\f_0)>_{\bT^k v} \lb{2.11}\ee
where
\be (\bT v)(\f) = \log\left[ \frac{\int P_0(dz) e^{v(\f+z)}} {\int P_0(dz)
e^{v(z)}} \right]^{\g^2} \lb{2.12} \ee
\be (\bL_v F)(\f) = \frac {\int P_0(dz) e^{v(\f+z)}F(\f+z)} {\int P_0(dz)
e^{v(\f+z)}} \lb{2.13} \ee
if $P_0(dz)$ denotes the gaussian measure on $\RR^1$ of mean zero
and covariance $\frac{1}{2\p}\log \g$.

The operators $\bT$ and $\bL_v$ appear also in the expressions similar to
\pref{2.11} valid for the expectations of any {\it observable} depending on
the values of the field $\f_x$ in a finite set of tesserae $\D\in Q_0$.

The operator \pref{2.12} is the {\it renormalization group transformation}.
It leaves invariant the space $\got{C}_\a$ of the continuous functions
periodic of period $T_\a=2\p/\a$ and satisfying \pref{2.4}; then, if we want
to study the hierarchical Coulomb gas at temperature $\b^{-1}$, we have to
restrict $\bT$ to $\got{C}_\a$ with $\a=\sqrt{\b e^2}$.  In Ref. \cite{BGN} it
was implicitly shown that $v_0(\f)=0$ \ is, for $\a^2>8\p$, a fixed point of
\pref{2.12} which is attracting for functions of the form \pref{2.6}, for $\l$
small enough.  In fact one could also show that it is locally attracting in
some subspace of sufficiently regular functions.

In this paper we shall study the more difficult case $\a^2\le 8\p$ and we
shall prove that there are two stable fixed points $v_\a(\f)\not=0$ (this
result, as discussed in the Introduction, has already been obtained with a
different technique \cite{MF}) in a suitable subspace of $\got{C}_\a$.

Moreover, by studying the spectrum of \pref{2.13} for $v=v_\a$ in the space
of $L_2$ functions periodic of period $T_\a$, we shall
prove that the integer charge truncated correlations decay like $d_{xy}^{-4}$.
The restriction of $\bL_v$ to periodic functions is motivated by the fact that
the truncated integer charge correlations are given by the formula
\be \r^T (x_1,\s_1;\ldots;x_n,\s_n) = \frac{\pd^n}{\pd\o_1 \cdots \pd\o_n}
\log < \exp\{\frac{\l}{2} \sum_{j=1}^n \o_j e^{i\a\s_j\f_{x_j}}\} > |_{\o_j=0}
\lb{2.14} \ee
where $\s_i\in \{-1,1\}$ are the charges and $x_i$ are the positions of $n$
particles.
%
%\input coul_3
%
\setcounter{equation}{0}
\section{The existence of two attracting nontrivial fixed points} \lb{s3}
The proof of the existence of nontrivial fixed points for $\a^2 < 8\p$
is based on the perturbative expansion of the renormalization group
transformation \pref{2.12}. If $v(\f)\in\got{C}_\a$, we can write:
\be v(\f)=\sum_{0\not=Q\in\msy{Z}} v_Q (e^{i\a Q\f}-1) \lb{3.1} \ee
where $v_Q=v_{-Q}$.
Then, if $v'_Q$ are the Fourier coefficients of $(\bT v)(\f)$, \pref{2.12} can
be written in the following form:
\be v'_Q = \g^{2-\frac{\a^2}{4\p}Q^2} v_Q + \g^2\sum_{n=2}^\infty \frac{1}{n!}
\sum_{Q_1+\cdots+Q_n=Q} v_{Q_1} \cdots v_{Q_n} F_n(Q_1,\ldots,Q_n) \lb{3.2}\ee
where
\be F_n(Q_1,\ldots,Q_n) = \ET(e^{i\a Q_1 z};\ldots;e^{i\a Q_n z}) \lb{3.3} \ee
if we denote $\ET$ the truncated expectation with respect to the measure
$P_0(dz)$.

In the appendix we prove the following nontrivial bound, which will play a
crucial role in the following:
\be |F_n(Q_1,\ldots,Q_n)| \le n^{n-1} \left[\prod_{i=1}^n\,(4\sqrt{\k}|Q_i|)
\right] e^{-\k\frac{Q^2}{n}} \lb{3.4} \ee
where $\k=(\a^2/8\p)\log \g$ and $Q=\sum_{i=1}^n Q_i$.

The linearization of \pref{3.2} around the fixed point $v=0$ has a bifurcation
at $\a^2= 8\p$, where the Fourier coefficients $v_{\pm 1}$ become unstable.
This bifurcation, as it is well known, is responsible for the
Kosterlitz-Thouless phase transition. In this paper we study the range of
temperatures, immediately above the critical one, given by the relation
\be 0 < \e\equiv 2-\frac{\a^2}{4\p} \le \e_0 \lb{3.5} \ee
with $\e_0$ small enough. We start by proving that \pref{3.2} has two fixed
points different from zero.

We first look for approximate solutions, by imposing in \pref{3.2} the
conditions $|Q|,|Q_i|\le 2$ and $n\le 2$. Taking into account the symmetry
property $v_Q=v_{-Q}$, we obtain a system of two equations:
\be \ba{rcl} v'_1 &=& \g^\e v_1 + a v_1 v_2 - f v_1^3 \asp
           v'_2 &=& c v_2 - b v_1^2 \ea \lb{3.6} \ee
where
\be \ba{rcl}
   a & = & \g^2 F_2(-1,2) = \g^\e (1-\g^{-4(2-\e)})\asp
   b & = & -\mezzo \g^2 F_2(1,1) = \mezzo \g^{-2(1-\e)} (1-\g^{-4+2\e})\asp
   c & = & \g^{-6+4\e}\asp
   f & = & -\mezzo \g^2 F_3(-1,1,1) = \mezzo \g^\e (1-\g^{-4+2\e})^2\\ & &
\ea \lb{3.7} \ee

\def\vbar{\bar v}
If $\e>0$ the system \pref{3.7} has two solutions different from zero given
by:
\be \ba{rcl}
 \vbar_1^2  & = &\dsty   \frac{(1-c)(\g^\e-1)}{ab+f(1-c) } \lasp
 \vbar_2   & = &\dsty   \frac{-b(\g^\e-1)}{ab+f(1-c)}  \ea \lb{3.8} \ee
Furthermore it is possible to see that, if $\bar v_1^{+}$ and $\bar v_1^{-}$
are, respectively, the positive and the negative solution, there is a
neighbourhood $\got{S}$ of the origin in $\RR^2$, containing $\bar v_1^{+}$
and $\bar v_1^{-}$, such that $\got{S}\cap\{v_1>0\}$ and
$\got{S}\cap\{v_1<0\}$ are in the domain of attraction, respectively, of $\bar
v_1^{+}$ and $\bar v_1^{-}$.


In the following we shall consider only the solution with $\vbar_1>0$, but
the same considerations could be applied to the other one. We want to prove
that there is a fixed point of \pref{2.12} which is {\it approximately equal}
to the function
\be \vbar(\f) = 2\vbar_1 [\cos\f-1] + 2\vbar_2 [\cos (2\f)-1] \lb{3.9}\ee
We want to apply the contraction mapping principle; hence we need to define a
suitable Banach space and find a $\bT$-invariant subset, containing
\pref{3.9}, on which $\bT$ is a contraction with respect to a suitable metric.

Let us consider the functions of $\got{C}_\a$, which are analytic and bounded
in a symmetric strip along the real axis of width $2\bt$ such that
\be \d  \equiv e^{-\a\bt} \equiv \bar a \e^\mezzo \;\le\; \bar\d\;<\; 1
\lb{3.10}\ee
These functions form a Banach space $\got{B}$, if we define the norm in the
following way:
\be ||v|| = \sup_{Q\ge 1} \d^{-Q} |v_Q| \lb{3.11} \ee
Let $\got{B}_d\subset \got{B}$ be the sphere of radius $d$ with center at the
origin.  We want to choose $d$ and $\bar a$, see \pref{3.10}, so that the
smaller sphere $\got{B}_{d/2}$ contains the function $\vbar$, see \pref{3.9}.
>From \pref{3.8} it follows that this is possible if
\be \ba{rcl} \dsty 2  \sqrt{\frac{1-c}{ab+f(1-c)}\frac{\g^\e-1}{\e}} & \le &
d\bar a\lasp \dsty 2  \frac{b}{ab+f(1-c)}\frac{\g^\e-1}{\e} & \le & d{\bar
a}^2 \ea \lb{3.12}\ee
The bounds \pref{3.12} and \pref{3.10} can be satisfied for any $d$, if $\bar
a$ is sufficiently large and $\e$ is sufficiently small, as we shall suppose in
the following.

We want now to define a subset $\got{D}\subset \got{B}_d$ containing the
functions which are {\it close} to $\vbar$. Let us define:
\be v_i = \vbar_i + r_i \virg v'_i=\vbar_i + r'_i \virg i=1,2 \lb{3.13}\ee
and consider the linear change of coordinates which diagonalizes the
linearization of the system \pref{3.6} around its fixed point, that is:
\be \left( \ba{c} r_1\\r_2 \ea \right) = S\; \left( \ba{c} u_1\\u_2 \ea
\right) \lb{3.14} \ee
where
\be S = \left( \ba{cc} 1 & -a\s\vbar_1\\ -2b\s\vbar_1 & 1 \ea \right)
\lb{3.15} \ee
with
\be \ba{rcl} \s &=& \dsty \frac{1}{1-c-K(\g^\e-1)}\\[15pt]
K & = & \dsty \frac{4(1-c)}{(1+2f\vbar_1^2-c) \left[ 1+\sqrt{1-
\frac{8(1-c)(\g^\e-1)}{(1+2f\vbar_1^2-c)^2}} \,\right]} \ea \lb{3.16}\ee
%
$\got{D}$ is the set of functions $v\in \got{B}_d$, such that:
\be \ba{rcl} |u_1| &\le& \dt \e^{1+\eta} \asp
|u_2| &\le& \dt \e^{3/2} \ea \lb{3.20}\ee
where $0< \eta <1/2$ and $\dt$ is any fixed positive constant.

\begin{theorem} \lb{Th. 3.1}
There exist a positive constant $d_0$ and, for any given $d,\dt,\eta$,
such that $0< d\le d_0$ and
$0<\eta<1/2$, another constant $\e_0$, so that the set $\got{D}$ is
invariant under the transformation $\bT$, if $\e\le\e_0$.
\end{theorem}
\Dim \pref{3.2} and \pref{3.4} imply that, if $v\in\got{B}_d$ and $Q\ge 1$:
\bea |v'_Q| &\le& \g^{2-Q^2(2-\e)} d\,\d^Q + \sum_{n=2}^\infty
(Bd\sqrt{\k})^n e^{-\k\frac{Q^2}{n}} \sum_{Q_1+\cdots+Q_n=Q} \prod_{i=1}^n
|Q_i| \d^{|Q_i|} \nn\\
&\le& \g^{2-Q^2(2-\e)} d\,\d^Q + \sum_{n=2}^\infty
(B_1d\sqrt{\k})^n e^{-\k\frac{Q^2}{n}} \sum_{Q_1+\cdots+Q_n=Q \atop |Q_i|\ge
1 } \prod_{i=1}^n \d_1^{|Q_i|} \lb{3.21} \eea
where $B,B_1$ are suitable positive constants  and $\d_1$ is chosen so that
\be \bar\d < \d_1 <1 \lb{3.22}\ee

We now have to carefully bound the sum:
\be I\equiv \sum_{n=2}^\infty
(B_1d\sqrt{\k})^n e^{-\k\frac{Q^2}{n}} \sum_{Q_1+\cdots+Q_n=Q \atop |Q_i|\ge
1} \prod_{i=1}^n \d_1^{|Q_i|} \lb{3.23} \ee
We can write:
\be I=I_0+I_1 \lb{3.24}\ee
with
\be I_0 = \sum_{n=2}^Q (B_1d\sqrt{\k})^n e^{-\k\frac{Q^2}{n}}
\d_1^Q N_n(Q) \lb{3.25} \ee
\be I_1 = \sum_{n=2}^\infty (B_1d\sqrt{\k})^n e^{-\k\frac{Q^2}{n}}
\sum_{k=1}^{n-1} \comb(n,k) \sum_{s=k}^\infty \d_1^{Q+2s} N_k(s)
N_{n-k}(Q+s) \lb{3.26} \ee
where the combinatorial factor $N_k(Q)$ is defined as
\be N_k(Q) = \sum_{Q_1+\cdots+Q_k=Q\atop Q_i\ge 1} \,1 = \cases{
\comb({Q-1},{k-1}) & for $Q\ge k\ge1$\cr &\cr\qquad 0 & for $1\le Q<k$ \cr }
\lb{3.27}\ee

It is easy to see that
\be I_0 \le \k Q e^{-\k Q} (B_1d)^2 (1+B_1d\sqrt{\k})^{Q-2} \d_1^Q \lb{3.28}\ee
In order to bound $I_1$, we use the inequality
\be \comb(s,k) \le \frac{s^k}{k!} \le \frac{e^{\r s}}{\r^k} \lb{3.29} \ee
valid for any positive $\r$ and we choose $\r$ so that
\be \d_2 \equiv \d_1 e^\r <1 \lb{3.30}\ee\
Then we have:
\be I_1 \le \d_2^Q (\d_1B_1d\sqrt{\k})^2 \sum_{n=2}^\infty \left(
\frac{B_1d\sqrt{\k}}{\r}\right)^{n-2} e^{-\k\frac{Q^2}{n}}
\frac{(1+\d_2^2)^n-1-\d_2^{2n}}{\d_2^2 (1-\d_2^2)} \lb{3.31}\ee
If $d$ is sufficiently small, we have also:
\be \frac{B_1d\sqrt{\k}}{\r} (1+\d_2^2) \le\bar\d< 1 \lb{3.32}\ee
which is compatible with \pref{3.10} and \pref{3.12} for $\e$ small enough and
$\bar a$ large enough. \pref{3.25}, \pref{3.31} and \pref{3.32} easily imply
that \be I \le \bar A (B_1d)^2 \d^Q \lb{3.33}\ee
for a suitable positive constant $\bar A$, only depending on $\bar\d$ and
$\d_1$.

The inequalities \pref{3.21} and \pref{3.33} imply that, if $Q\ge 3$,
$|v'_Q|\le d\d^Q$ provided that
\be \g^{2-9(2-\e)} d + \bar A (B_1d)^2 \le d \lb{3.34}\ee
which can be satisfied if $d$ is sufficiently small. Then, in order to prove
that \got{D} is invariant, we still have to check that the condition
\pref{3.20} is preserved.

Let us notice that:
\be \comb({r'_1},{r'_2}) = \left( \ba{cc} 1-2f\vbar_1^2 & a\vbar_1\\
-2b\vbar_1 & c \ea \right)
\comb({r_1},{r_2}) + \comb({\psi_1},{\psi_2}) + \comb({\vt_1},{\vt_2})
\lb{3.35}\ee where
\be \ba{rcl}  \psi_1 &=& ar_1r_2 -3f\vbar_1 r_1^2 -fr_1^3\asp \psi_2 &=&
-br_1^2 \ea \lb{3.36}\ee
and
\be \ba{rcl} \vt_1 &=&
\dsty \g^2\sum_{n=2}^\infty \frac{1}{n!}
\sum_{Q_1+\cdots+Q_n=1\atop |Q_1|+\cdots |Q_n|\ge 5} v_{Q_1} \cdots v_{Q_n}
F_n(Q_1,\ldots,Q_n) \lasp
\vt_2 &=&\dsty  \g^2\sum_{n=2}^\infty \frac{1}{n!}
\sum_{Q_1+\cdots+Q_n=2\atop |Q_1|+\cdots |Q_n|\ge 4} v_{Q_1} \cdots v_{Q_n}
F_n(Q_1,\ldots,Q_n) \ea \lb{3.37}\ee

Proceeding as before, it is easy to show that, if $d$ is sufficiently small,
then
\be \ba{rcl} |\vt_1| &\le& d\d^5 = d\bar a^5 \e^{5/2} \asp
|\vt_2| &\le& d\d^4 = d\bar a^4 \e^2 \ea \lb{3.38}\ee

The previous considerations imply that there exists $d_0>0$, such that, given
$d\le d_0$, \pref{3.10}, \pref{3.12}, \pref{3.32}, \pref{3.34} and \pref{3.38}
are satisfied for $\e$ sufficiently small.

By some simple algebra and using \pref{3.14}, it is possible to show that
\be \left( \ba{c} u'_1\\u'_2 \ea \right) = \left( \ba{c} \l_1u_1\\\l_2u_2 \ea
\right) + \left( \ba{c} R_1\\R_2 \ea \right) +
\comb( {\vt_1 +a\tilde\s\vbar_1\vt_2},
{\vt_2 +2b\tilde\s\vbar_1\vt_1}) \lb{3.17} \ee
where
\be \ba{rcl} \l_1 & = & 1- K(\g^\e-1) \asp \l_2 & = & \g^{-6+4\e} + K(\g^\e-1)
-2f\vbar_1^2 \ea \lb{3.18}\ee
and
\be \left( \ba{c} R_1\asp R_2 \ea \right) = \tilde\s \left( \ba{c}
ar_1r_2- (3f+ab\s)\vbar_1 r_1^2 -fr_1^3 \asp
-br_1^2 +2b\tilde\s \vbar_1(ar_1r_2-3f\vbar_1r_1^2-fr_1^3) \ea \right)
\lb{3.19}\ee
\be \tilde\s = \frac{\s}{1-2ab\s^2 \vbar_1^2} \lb{3.19a}\ee

By using \pref{3.8}, \pref{3.14}, \pref{3.38}, and \pref{3.17}, it is easy to
prove that, if $0<\eta<1/2$, and $\e$ is sufficiently small, say $\e\le\e_0$,
then
\be \ba{rcl} |u'_1| &\le& \l_1 \dt \e^{1+\eta} + c_1 \dt^3 \e^{5/2+\eta} +
c_2 d \bar a^5 \e^{5/2} \asp
|u'_2| &\le& \l_2 \dt \e^{3/2} + c_3 \dt^3 \e^{2+2\eta} +c_4 d \bar a^5 \e^2
\ea \lb{3.40}\ee
for suitable constants $c_i, i=1,\ldots,4$. Then the conditions \pref{3.20}
are preserved if
\be \ba{rcl} \dsty \l_1 + c_1 \dt^2 \e^{3/2} + c_2\frac{d}{\dt} \bar a^5
\e^{3/2-\eta} &\le&
1\asp \dsty \l_2 + c_3 \dt^2 \e^{1/2+2\eta} + c_4 \frac{d}{\dt} {\bar a}^5
\e^{1/2} &\le& 1 \ea \lb{3.41}\ee
By looking at \pref{3.18}, it is immediate to see that \pref{3.41} can be
satisfied, given any $0<\eta<1/2$, if $\e$ is small enough. In order to prove
that $\bT v\in\got{D}$, we still have to check that
\be |v'_1|\le d\d \virg |v'_2|\le d\d^2 \lb{3.42}\ee
which is again true for any $\eta<1/2$, if $\e$ is small enough, by
\pref{3.13}, \pref{3.20} and the fact that ${\bar v}\in\got{B}_{d/2}$. \Halmos

We now want to show that $\got{D}$ contains a fixed point $v^*$ of the
transformation $\bT$ and that, given any $v\in\got{D}$, $||\bT^n v-v^*||\to 0$
as $n\to\infty$.

Given two elements of $\got{D}$, $v^{(1)}$ and $v^{(2)}$, we define
$r^{(j)}_i= v^{(j)}_i -\bar v^{(j)}_i$, $j,i=1,2$, and $u^{(j)}_i$ as in
\pref{3.13} and \pref{3.14}; then we define:
\be \got{m} (v^{(1)},v^{(2)}) =\max \{ \d^{-1} |u^{(1)}_1 -u^{(2)}_1|,
\d^{-2} |u^{(1)}_2 -u^{(2)}_2|, \sup_{Q\ge 3} \d^{-Q} |v^{(1)}_Q
-v^{(2)}_Q| \} \lb{3.43}\ee

It is easy to see that
\be \ba{l} |v^{(1)}_1 -v^{(2)}_1| = |r^{(1)}_1 -r^{(2)}_1| \le c_5\,\d\,
\got{m}( v^{(1)}, v^{(2)})\asp
|v^{(1)}_2 -v^{(2)}_2| = |r^{(1)}_2 -r^{(2)}_2| \le c_5\, \d^2 \got{m}(
v^{(1)}, v^{(2)}) \ea \lb{3.44}\ee
and
\be \ba{l} |u^{(1)}_1 -u^{(2)}_1| \le c_5\, \d\; || v^{(1)} - v^{(2)}|| \asp
|u^{(1)}_2 -u^{(2)}_2| \le c_5\, \d^2\, || v^{(1)} - v^{(2)}|| \ea
\lb{3.45}\ee for some constant $c_5$.

The inequalities \pref{3.44} and \pref{3.45} imply that $\got{m}$ and the norm
\pref{3.11} generate the same topology. Therefore, in order to prove the
existence of a fixed point $v^*$ in $\got{D}$ and its asymptotic stability,
the following theorem is sufficient.

\begin{theorem} \lb{Th. 3.2}
There exist a positive constant $d_1\le d_0$ and, for any given $d\le d_1$ and
$0<\eta<1/2$, another constant $\e_1\le\e_0$, such that, for any $\e\le\e_1$:
\be \got{m}(\bT v^{(1)},\bT v^{(2)}) \le \n_\e \got{m}(v^{(1)},v^{(2)})
\lb{3.46}\ee with $\n_\e < 1$ and $\n_\e \to 1$ as $\e\to 0$.
\end{theorem}
\Dim By proceeding as in the proof of Theorem \ref{Th. 3.1} and using the
identity
\be v^{(1)}_{Q_1} \cdots v^{(1)}_{Q_n} - v^{(2)}_{Q_1} \cdots v^{(2)}_{Q_n}
= \sum_{k=1}^n v^{(1)}_{Q_1} \cdots v^{(1)}_{Q_{k-1}} [ v^{(1)}_{Q_k} -
v^{(2)}_{Q_k} ] v^{(2)}_{Q_{k+1}} \cdots v^{(2)}_{Q_n} \lb{3.47}\ee
and \pref{3.44}, it is easy to show that, if $Q\ge 3$ and $d$ is sufficiently
small, then
\be \d^{-Q} |(\bT v^{(1)})_Q - (\bT v^{(2)})_Q | \le [\g^{2-9(2-\e)} + B_2 d]
\got{m} (v^{(1)},v^{(2)}) \lb{3.48}\ee
for some constant $B_2$, depending only on $\bar\d$ and $\d_1$.

Moreover, by using \pref{3.17}, it is easy to show that
\be \ba{rcl}
\d^{-1} |u'^{(1)}_1 - u'^{(2)}_1 | &\le& [\l_1 + c_6 \e^{3/2} + c_6
\e^2]\, \got{m} (v^{(1)},v^{(2)})  \asp
\d^{-2} |u'^{(1)}_2 - u'^{(2)}_2 | &\le& [\l_2 + c_6 \e^{1/2+\eta} + c_6
\e ] \, \got{m} (v^{(1)},v^{(2)}) \ea \lb{3.49}\ee
for $d$ small enough, say $d\le d_1\le d_0$, and some constant $c_6$ depending
on $\dt,\bar a$ and $d_1$.

All the claims of the theorem easily follow. \Halmos

Theorem \ref{Th. 3.2} is not completely satisfactory, since we are
interested in the properties of the measure \pref{2.5} with potential $v(\f)=
\l(\cos \f -1)\equiv v^{(\l)}(\f)$. The properties of the approximate
transformation \pref{3.6} (see discussion after \pref{3.8}) suggest that
$v^{(\l)}$ is in the domain of attraction of $v^*$, for $\l$ positive and
sufficiently small, and that
a similar result holds for $\l<0$; moreover, the computer simulation is in
complete agreement with this conjecture. In order to rigorously prove this
claim, however, we should investigate more accurately the properties of the
equation \pref{3.17}, trying to show that $\bT^n v^{(\l)}\in\got{D}$ for any
$\l$ sufficiently small, if $n$ is large enough. We think that this is
possible, but we did not try to fill in the details.

%
%\input coul_4
%
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\section{The correlation functions} \lb{s4}
Let us suppose that $\e$, $\eta$ and $d$ are chosen so that there is in
$\got{D}$ a fixed point $v^*$ of the transformation $\bT$. We want to study
the linear operator $\bL_v$, see \pref{2.13}, for $v=v^*$; let us simply call
it $\bL$.

We shall consider the action of $\bL$ on the Hilbert space $\got{H}$ of the
functions periodic of period $T_\a=2\p/\a$ with inner product
\be (G,F) = \frac{1}{T_\a} \int_0^{T_\a} d\f\ q(\f) G^*(\f) F(\f) \lb{4.1}\ee
where
\be q(\f)= e^{v(\f)} \int P_0(dz) e^{v(\f+z)} \equiv e^{v(\f)} N(\f)
\lb{4.2}\ee

\begin{prop} \lb{p4.1}
$\bL$ is a trace class positive self-adjoint operator of norm $1$.
\end{prop}
\Dim It is very easy to verify that $\bL$ is self-adjoint, by using the
fact that the measure $P_0(dz)$ is even in $z$.

Let us now observe that the functions
\be \psi_Q(\f) = q(\f)^{-\mezzo} e^{i\a Q\f} \virg Q\in\msy{Z} \lb{4.3}\ee
are a base of $\got{H}$ and that we can write:
\be e^{v(\f)} q(\f)^{-\mezzo} = \sum_Q g_Q e^{i\a Q\f} \virg \sum_Q |g_Q|^2
<\infty \lb{4.4}\ee
Then we have:
\be \ba{l}
Tr(\bL) = \sum_Q (\psi_Q,\bL\psi_Q) =\lasp
\dsty\quad =  \sum_Q \int P_0(dz)
\frac{1}{T_\a} \int_0^{T_\a} d\f [ e^{v(\f)} \psi_Q^*(\f)] [ e^{v(\f+z)}
\psi_Q(\f+z)] =\lasp
\dsty\quad = \sum_{Q,Q'} |g_{Q'-Q}|^2 \int P_0(dz) e^{i\a Q' z}=
\sum_{Q,Q'} |g_{Q'-Q}|^2 \g^{-\frac{\a^2}{4\p}Q'^2} =\lasp
\dsty\quad = (\sum_{Q} |g_Q|^2)\; (\sum_{Q'} \g^{-\frac{\a^2}{4\p}Q'^2}) <
\infty \ea \lb{4.5}\ee
which proves that $\bL$ is trace class.

If $F\in\got{H}$, $e^vF\in \LL_2$ and therefore we can write
\be e^{v(\f)} F(\f) = \sum_Q {\tilde f}_Q e^{i\a Q\f} \virg \sum_Q |{\tilde f}
_Q|^2 <\infty \lb{4.6}\ee
Then, by proceeding as before, we can check that
\be (F,\bL F) = \sum_Q |\tilde f_Q|^2 \g^{-\frac{\a^2}{4\p}Q^2} \lb{4.7}\ee
which proves that $\bL$ is a positive operator. Moreover
\be \ba{l} \dsty (F,\bL F) = |(F,\bL F)| \le\lasp
\dsty \le \int P_0(dz)
\frac{1}{2T_\a} \int_0^{T_\a} d\f e^{v(\f) + v(\f+z)} [|F(\f)|^2 +
|F(\f+z)|^2] =\lasp
\dsty = \frac{1}{T_\a} \int_0^{T_\a} d\f e^{v(\f)} |F(\f)|^2 N_v(\f) =
(F,F) \ea \lb{4.8}\ee
Since $\bL F=F$, if $F$ is a constant function, $||\bL||=1$. \Halmos

By Prop. \ref{p4.1} $\bL$ is a positive compact operator, then it has a pure
discrete point spectrum with positive eigenvalues, at most finitely
degenerate. Furthermore the subspaces $\got{H}^+$ and $\got{H}^-$ of
$\got{H}$, which contain the functions even and odd in $\f$ respectively, are
invariant under the action of $\bL$. Let $\bL^\pm$ be the restriction of $\bL$
to $\got{H}^\pm$.

For $\e=0$ (and hence $v=0$) the eigenvalues of $\bL^+$ and $\bL^-$ are the
same, that is:
\be \l_n^\pm(0) = \g^{-2n^2} \virg n=0,1,\ldots \lb{4.9}\ee
and they are all simple. By using the properties of $v$ proven in Sect.
\ref{s3} and known results about the perturbation theory of compact operators,
it is possible to show that the eigenvalues of $\bL^\pm$ can be written as
suitable functions $\l_n^\pm(\e)$, which are continuous in $\e=0$.

Since the constants are eigenfunctions of $\bL$ for any $\e\le\e_1$:
\be \l_0^+ = \l_0^- =1 \lb{4.10}\ee
and $1$ is a simple eigenvalue of $\bL$. All the other eigenvalues are
strictly less than 1.

Let $\{\m_n\}_{n\ge0}$ be the set of all eigenvalues, ordered so that
$\m_{n+1} \le \m_n$, and let $G_n$ the corresponding eigenfunctions,
normalized so that $G_n$ is real and
\be (G_n, G_m) = \d_{nm} \lb{4.11}\ee
In particular $\m_0 =1$ and
\be G_0 = \left[ \frac{1}{T_\a} \int_0^{T_\a} d\f q(\f) \right]^{-\mezzo} \ee
Moreover, it is possible to show, by the technique used below in the proof of
Theorem \ref{Th. 4.1}, that the $G_n$ are smooth functions.

Let us consider a function $F\in\got{H}$ such that its expansion
in terms of the $G_n$:
\be F(\f) = \sum_{n=0}^\infty f_n G_n(\f) \lb{4.11a}\ee
has good convergence properties. Then, by \pref{2.11} we have:
\be \ba{l} <F(\f_0)>_v = <(\bL^k F)(\f_0)>_v =\asp
\dsty \quad = \sum_n f_n \m_n^k <G_n(\f_0)>_v \tende{k\to\infty} f_0 G_0 =
(G_0,F)G_0 \ea \lb{4.12}\ee

Let us now suppose that we want to calculate the correlation between
$F(\f_0)$ and $F^*(\f_x)$. This problem arises, for example, if we are
interested in the two charges correlation; in this case
$F(\f)=e^{i\a\f}$, whose expansion has the needed convergence properties, as
it is possible to show with some standard calculation, using the smoothness of
the functions $G_n$ and the fact that they are small perturbations of the
functions $e^{i\a Q}$.

If $h$ is the smallest integer so that there exists a $\D\in Q_h$ containing
both $0$ and $x$, we can write:
\be \ba{l} <F(\f_0) F^*(\f_x)>_v = <|(\bT^hF)(\f_0)|^2>_v =\asp \dsty
\quad = \sum_{nm} f_n f_m \m_n^h \m_m^h <G_n(\f_0) G_m(\f_0)>_v
\ea \lb{4.13}\ee
But, by \pref{4.11} and \pref{4.12}:
\be <G_n(\f_0) G_m(\f_0)>_v = (G_0,G_nG_m)G_0 = G_0^2 (G_n,G_m) = G_0^2
\d_{nm} \ee
Then
\be \ba{rcl} <F(\f_0) F^*(\f_x)>_v^T &=& <F(\f_0) F^*(\f_x)>_v -
|<F(\f_0)>_v|^2
= \asp &=& \dsty G_0^2 \sum_{n=1}^\infty \m_n^{2h} |f_n|^2 \ea \ee
and, as a consequence
\be <F(\f_0) F^*(\f_x)>_v^T \;{\raisebox{-.5ex}{$\simeq$}\atop
\raisebox{.5ex}{$x\to\infty$}} \;c \m_1^{2h} = c d_{0x}^{-\t} \ee
with $c$ a suitable constant and
\be \t = -2\log_\g \m_1 \lb{4.15}\ee

We now want to show that
\be \m_1 = \g^{-2} \lb{4.16}\ee
at least for $\e$ small enough. We start by observing that, since $\bT v=v$,
by \pref{2.12}:
\be e^{\frac{1}{\g^2} v(\f)} = \frac{\int P_0(dz) e^{v(\f+z)}}
{\int P_0(dz) e^{v(z)}} \lb{4.17} \ee
By calculating the $\f$-derivative of both sides, it is easy to check that:
\be \bL \frac{dv}{d\f} = \g^{-2} \frac{dv}{d\f}\ee
Since $\frac{dv}{d\f}\in \got{H}^-$, this implies that
\be \l_1^-(\e) = \g^{-2} \virg \forall \e\le \e_1 \lb{4.19}\ee
But $\m_1$, for $\e$ small enough, is the minimum between $\l_1^+(\e)$
and $\l_1^-(\e)$; hence, in order to show \pref{4.16} it is sufficient to
prove that $\l_1^+(\e)$ is smaller than $\g^{-2}$.

We notice that \pref{2.13} can be written also in the following way:
\be (\bL_v F)(\f) = \frac{d}{d\l} \left. \log \int P_0(dz) e^{v(\f+z) + \l
F(\f+z)} \right|_{\l=0} \lb{4.20} \ee

If $F\in\got{H}$, $F\in\LL_2$, then we can write:
\be F(\f) = \sum_Q f_Q e^{i\a Q\f} \ee
Moreover, since $(\bL_v F)(\f)$ does not change if we add a constant to
$v(\f)$ , we can replace in \pref{4.20} $v(\f)$ by the expansion
$\sum_{Q\not=0} v_Q e^{i\a Q\f}$, whose coefficients are the same as in
\pref{3.1}. Then we obtain:
\be \ba{l} \dsty (\bL_v F)(\f) =  \sum_Q f_Q \g^{-\frac{\a^2}{4\p}Q^2}
e^{i\a Q\f} +\lasp
\dsty +\sum_{n=2}^\infty \frac{1}{(n-1)!} \sum_{Q_1,\ldots,Q_n} v_{Q_1}
\cdots v_{Q_{n-1}} f_{Q_n} F_n(Q_1,\ldots,Q_n) e^{i\a (\sum_{i=1}^n Q_i)\f}
\ea \lb{4.21}\ee
If $\l<1$, the eigenvalue equation $\bL_v F=\l F$ is satisfied if
$f_0=0$ and
\be \ba{rcl} \l f_Q &=&  f_Q \g^{-\frac{\a^2}{4\p}Q^2} +\lasp
&+& \sum_{n=2}^\infty \frac{1}{(n-1)!} \sum_{Q_1+\cdots+Q_n=Q} v_{Q_1} \cdots
v_{Q_{n-1}} f_{Q_n} F_n(Q_1,\ldots,Q_n) \ea \lb{4.22}\ee
for $|Q|\not=0$.

We are interested in the dependence on $\e$ of the eigenvalue $\l_1^+(\e)$ and
of the Fourier coefficients $f_Q(\e)=f_{-Q}(\e)$ of the corresponding
eigenfunction, which we shall normalize so that
\be f_1(\e)=f_{-1}(\e)=1 \lb{4.22a}\ee
For $\e=0$, we have
\be f_1(0)=f_{-1}(0)=1 \virg f_Q=0 \quad \mbox{if} |Q| \not=1 \ee

We shall now rewrite \pref{4.22}, for $\l=\l_1^+(\e)$, as a fixed point
equation in a suitable Banach space, where the existence of a unique solution
will follow from the contraction mapping principle.

Let us define:
\be \ba{l} \dsty G_Q(F) = \sum_{n=2}^\infty \frac{1}{(n-1)!} \cdot \lasp
\dsty \quad \cdot \sum_{|Q'|\ge 2} f_{Q'} \sum_{Q_1+\cdots+Q_{n-1}=Q-Q'}
v_{Q_1} \cdots v_{Q_{n-1}} F_n(Q_1,\ldots,Q_{n-1},Q') \ea\lb{4.27}\ee
Eq. \pref{4.22} gives, for $Q=1$:
\be \l_1^+(\e) = \g^{-2} + r(\e) + G_1(F) \lb{4.25}\ee
where
\be \ba{l} \dsty r(\e) = \g^{-2}(\g^\e -1) + v_2 F_2(2,-1) +
 \sum_{n=3}^\infty \frac{1}{(n-1)!} \cdot\lasp
\dsty \cdot \quad \sum_{Q=\pm 1} \sum_{Q_1+\cdots+Q_{n-1}=1-Q} v_{Q_1}
\cdots v_{Q_{n-1}} F_n(Q_1,\ldots,Q_{n-1},Q) \ea \lb{4.26}\ee
If $|Q|\ge 2$, we have:
\be {[}\l_1^+(\e) - \g^{-\frac{\a^2}{4\p}Q^2}] f_Q = h_Q + G_Q(F) \lb{4.30}\ee
where
\be h_Q = \sum_{n=2}^\infty \frac{1}{(n-1)!}
\sum_{|Q'|=1} \sum_{Q_1+\cdots+Q_{n-1}=Q-Q'} v_{Q_1}
\cdots v_{Q_{n-1}} F_n(Q_1,\ldots,Q_{n-1},Q') \lb{4.31}\ee

By proceeding as in the proof of Theorem \ref{Th. 3.1}, it is easy to show
that there exist constants $a_1$ and $A_1$, only depending on $\bar \d$ and
$\d_1$, such that
\be |r(\e)| \le a_1\e \lb{4.32}\ee
\be |h_Q| \le A_1 d \d^{|Q|-1} \virg \mbox{if $|Q|\ge 2$} \lb{4.33}\ee

Let us now consider the Banach space $\got{E}$ of the even functions
$F(\f)$, periodic of period $T_\a$, such that $f_0=0$, $f_1=f_{-1}=1$ and
\be ||F||\equiv \sup_{Q\ge 2} \d^{-Q+1} |f_Q| < \infty \ee
We denote $\got{E}_D$ the sphere of radius $D$ and center at the origin
and we consider the operator $\bK$ from $\got{E}_D$ to $\got{H}_+$,
defined so that, if $(\bK F)(\f)= \sum_{Q\not=0} f'_Q e^{i\a Q\f}$, then
\be \ba{rcl} f'_Q=f'_{-Q} &=& \dsty \frac{h_Q + G_Q(F)}{\g^{-2} + r(\e) +
G_1(F) - \g^{-\frac{\a^2}{4\p}Q^2} } \virg \mbox{if $Q\ge 2$} \\[12pt]
f'_1=f'_{-1} &=& 1 \virg f'_0=0 \ea \lb{4.33a}\ee
By \pref{4.25} and \pref{4.30} $F$ is a solution of \pref{4.22} for
$\l=\l_1^+(\e)$, belonging to $\got{E}_D$, if and
only if $F$ is a fixed point of the operator $\bK$.

\begin{theorem} \lb{Th. 4.1}
There exist a positive constant $d_2\le d_1$ and, for any given $d\le d_2$ and
$0<\eta<1/2$, other constants $\e_2$, $D_0$ and $D_1$, such that $\got{E}_D$
is invariant under the transformation $\bK$, if $\e\le\e_2$ and $D_0 \le D\le
D_1$; moreover $\bK$ is a contraction as an operator from  $\got{E}_D$ to
$\got{E}_D$.
\end{theorem}

\Dim
By proceeding as in the proof of Theorem \ref{Th. 3.1}, it is possible to
show that, if $Q\ge 2$ and $Dd$ is small enough:
\be |f'_Q| \le \frac{A_1 d(1+D)\d^{Q-1}}{\g^{-2} -a_1\e -
DdA_1\d - \g^{-\frac{\a^2}{\p}} } \ee
Therefore $\got{E}_D$ is invariant under the transformation $\bK$ if
\be D \ge \frac{A_1 d(1+D)}{\g^{-2} -a_1\e -
DdA_1\d - \g^{-\frac{\a^2}{\p}} } \lb{4.35}\ee
and it is easy to see that there exist $d_2$, $\e_2$, $D_0$ and $D_1$ such
that \pref{4.35} is satisfied if $d\le d_2$, $\e\le\e_2$ and $D_0\le D\le
D_1$.

Let us now consider two elements $F_1,F_2\in\got{E}_D$ such that, for $Q\ge
2$,
\be |f_{1Q}-f_{2Q}| \le \r\d^{Q-1} \ee
It is easy to check that, for any $Q\ge 1$:
\be |G_Q(F_1)-G_Q(F_2)| \le \r d A_1\d^{Q-1} \lb{4.36}\ee
Moreover, if $Q\ge 2$, by \pref{4.33a}:
\be \ba{l} f'_{1Q} -f'_{2Q} = \{ h_Q [G_1(F_2)-G_1(F_1)] +b_Q
[G_Q(F_1)-G_Q(F_2)] +\lasp
\quad + G_Q(F_1) [G_1(F_2)-G_1(F_1)] + G_1(F_1) [G_Q(F_1)-G_Q(F_2)] \} /\lasp
\quad / \{ [b_Q+G_1(F_1)][b_Q+G_1(F_2)] \} \ea \ee
where
\be b_Q= r(\e)+\g^{-2}- \g^{-\frac{\a^2}{4\p}Q^2}\ee
Then it is very easy to show that, for $d$ small enough,
\be |f'_{1Q} -f'_{2Q}| \le {\bar\a} \r \virg {\bar\a}<1\ee
which immediately implies, together with Theorem \ref{Th. 3.1}, all the claims
of this theorem. \Halmos

>From \pref{4.27}, \pref{4.25}, \pref{4.26}, Theorem \ref{Th. 4.1} and some
simple algebra follows that:
\be \l^+_1(\e) = \g^{-2} - \e \log\g (1+\frac{ab}{ab+f(1-c)}) + O(\e^{3/2})\ee
Then, if $\e$ is small enough
\be \l^+_1(\e) < \l^-_1(\e) \ee
so that \pref{4.16} is satisfied.

This means that, if $v=v^*$, the integer charge truncated correlations decay
as $d_{xy}^{-4}$. With some more computational effort one could show that this
result is true also if $v$ is in the domain of attraction of $v^*$.

We conclude by two remarks. The first remark, anticipated in the discussion
preceding \pref{4.11a}, is that the technique used in this section can be
applied to any eigenvalue of $\bL_v$ with similar results. In particular, one
can show that, for any given $n$ and $\e$ small enough (how small depends on
$n$), $\l_n^{\pm}(\e) < \l_n^{\pm}(0)$.

The second remark is that we could study the fractional charges correlations,
by using similar arguments, in spite of the fact that the function $e^{i\a\f}$
must be substituted by $e^{i\a\xi\f}$, $0< \xi <1$, which is not periodic of
period $T_\a$. It is only sufficient to observe that, if $\bar F(\f)=
e^{i\a\xi\f} F(\f)$, with $F\in\got{H}$, then
\be (\bL_v \bar F)(\f) = e^{i\a\xi\f} (\bL_v^{(\xi)} F)(\f) \ee
\be (\bL_v^{(\xi)} F)(\f) = \frac {\int P_0(dz) e^{i\a\xi\f} e^{v(\f+z)}F(\f+z)}
{\int P_0(dz) e^{v(\f+z)}}  \ee
and $\bL_v^{(\xi)}$ is again a self-adjoint operator from $\got{H}$ to
$\got{H}$, whose spectrum can be studied in the same way as the spectrum of
$\bL_v$, obtaining the same results reported in Ref. \cite{MF}.

\vspace{1cm}

\noindent {\bf Acknowledgements.}\\ We are indebted to G. Gallavotti
and F. Nicol\`o for many discussions and suggestions.
\vglue1cm

%
%\input a
%
\def\cd{{c\over 2}} \def\cq{{c\over 4}} \def\cn{{c\over 4n}}
\def\JI{{\emptyset\not= J\msy{\$} I}} \def\sqc{\sqrt{c}}
\def\bs{\backslash}
\appendix\section{Proof of the bound (3.4)}
\setcounter{equation}{0}
\renewcommand{\theequation}{A\arabic{equation}}

Let $I=\{1,\dots,n\}$ be the set of the first $n$ positive integers and, for
each $i\in I$, let $Q_i$ be a fixed integer ($Q_i\in {\bf Z}$). If $z_t$ is a
random gaussian variable with mean $0$ and covariance
\be \E(z_t^2)=t\le 1 \ee
and $c$ is a fixed positive constant, we define:
\be F(I,t)=\ET(e^{i\sqc Q_1z_t};\dots;e^{i\sqc Q_nz_t}) \ee
where $\ET$ denotes the truncated expectation with respect to $z_t$. It is a
well known fact that:
\be F(I,t) = e^{-\cd\sum_{i=1}^n Q_i^2 t} f(I,t) \ee
with
\be f(I,t) = \sum_G \prod_{ij\in G} (e^{-cQ_iQ_jt}-1) \ee
where $G$ is the family of all connected graphs on $n$ vertices labeled by the
elements of $I$, with bonds denoted by $ij (i,j\in I)$.

Using the results of Ref. \cite{BK}, in particular Lemma 3.3 and the
recurrence
relation for $f(I,t)$ which follows from it (see pag. 40 of \cite{BK}), it is
very easy to show that:
\be F(I,t)= \left\{
\ba{l} \dsty
-\cd\int_0^t ds e^{-\cd Q_I^2(t-s)} \sum_\JI Q_JQ_{I\bs J}F(J,s)F(I\bs J,s)
\lasp
\mbox{\hspace{7cm} if $|I|=n\ge 2$} \lasp
\dsty e^{-\cd Q_I^2t} \mbox{\hspace{5.7cm} if $|I|=1$} \ea
\right. \lb{a5} \ee
where we used the definition, for $J$ a subset of $I$:
\be Q_J = \sum_{i\in J} Q_i \ee

We want now to describe the solution of the recurrence relation \pref{a5}. Let
us consider, for $n\ge 2$, the family $\G_n$ of all planar binary trees with
root $r$ and $n$ endpoints labeled by the elements of $I$, oriented from the
root to the endpoints, see Fig. 1.

\begin{figure}[hbt]
\begin{center}
%
%\input fig1
%
\begin{picture}(300,150)
\put(20,70){\line(1,0){50}}
\put(15,60){\makebox(0,0){$r$}}
\put(65,60){\makebox(0,0){$v_0$}}
\put(70,70){\line(3,2){30}}
\put(70,70){\line(2,-1){80}}
\put(100,90){\line(3,1){60}}
\put(155,117){\makebox(0,0){$v'$}}
\put(100,90){\line(6,-1){160}}
\put(160,110){\line(3,-1){100}}
\put(160,110){\line(2,1){40}}
\put(195,135){\makebox(0,0){$v$}}
\put(200,130){\line(5,1){60}}
\put(200,130){\line(5,-1){60}}
\put(150,30){\line(6,1){110}}
\put(150,30){\line(6,-1){60}}
\put(210,20){\line(6,1){50}}
\put(210,20){\line(6,-1){50}}
\end{picture}

Fig. 1
\end{center}
\end{figure}

We call vertices the root, the endpoints ($e.p.$ in the following) and the
branch points of the tree;
the branch points will be called also nontrivial ($n.t.$ in the
following) vertices. If $v$ is a vertex different from $r$, we shall denote by
$v'$ the vertex immediately preceding it in the tree and we shall say that
$i\in v$ if the $e.p.$ with label $i$ follows $v$; moreover
$v_0$ will denote the vertex immediately following the root.
We define:
\be Q_v = \sum_{i\in v} Q_i \ee
Finally we label each vertex $v$ with a real number $s_v$ such that:
\be \ba{c}
t\ge s_{v'}\ge s_v\ge 0 \\
s_r=t  \\
s_v=0 \quad \mbox{if $v$ is an $e.p.$} \ea\lb{a8} \ee

It is easy to see that, if $|I|\ge 2$:
\be F(I,t)=\sum_{\th\in\G_n} \int \left( \prod_{v \,n.t.} {-ds_v\over 2}
\right) \chi_\th W_\th \ee
where $\chi_\th$ is the characteristic function of the set \pref{a8} and
\be \ba{rcl} W_\th &=& \dsty \prod_{i=1}^n (\sqc Q_i)\left(\prod_{v\,e.p.}
e^{-\cd Q_i^2s_{v'}}\right) \cdot \lasp
&\cdot& \dsty \left[ \prod_{{v\,n.t.\atop v\not= v_0}}
(\sqc Q_v)e^{-\cd Q_v^2(s_{v'}-s_v)} \right]e^{-\cd Q_I^2(t-s_{v_0})} \ea
\lb{a10} \ee

We want to show that
\be |F(I,t)| \le n^{n-1} \prod_{i=1}^n (2\sqc |Q_i|) e^{-\cn Q_I^2t} \lb{a11}
\ee
 The first step in the proof is to get rid of the ``bad'' factors $Q_v$ in
\pref{a10}, using the bound
\be \sqc |Q_v|e^{-\cd Q_v^2(s_{v'}-s_v)} \le {1\over \sqrt{s_{v'}-s_v}}
e^{-\cq Q_v^2(s_{v'}-s_v)} \ee
Then, if $|I|\ge 2$, we can write, using also that $0\le t-s_{v_0}\le 1$:
\be |F(I,t)| \le \prod_{i=1}^n (\sqc |Q_i|) E(I,t) \lb{a14} \ee
where
\be E(I,t)= \left\{ \ba{l}
\dsty {1\over 2}\int_0^t {ds\over \sqrt{t-s}} e^{-\cq Q_I^2(t-s)}
\sum_\JI E(J,s)E(I\bs J,s) \lasp
\mbox{\hspace{7cm} if $|I|=n\ge 2$} \lasp
\dsty e^{-\cd Q_I^2t} \mbox{\hspace{5.7cm} if $|I|=1$} \ea
\right. \lb{a15} \ee

We now prove, by induction on $n=|I|$, that:
\be E(I,t) \le (2n)^{n-1}e^{-\cn Q_I^2t} \lb{a16} \ee
In fact \pref{a16} is true for $n=1$; moreover, if we suppose that it is true
for $1\le k <n$, we have, using \pref{a15}:
\be E(I,t) \le 2^{n-3}\sum_\JI k^{k-1}(n-k)^{n-k-1} G(I,J,t) \lb{a17} \ee
where $k=|J|$ and
\be G(I,J,t)= e^{-\cq Q_I^2t}\int_0^t {ds\over \sqrt{t-s}}
e^{\cq s[Q_I^2-Q_J^2/k-Q_{I\bs J}^2/(n-k)]} \ee
If $[Q_I^2-Q_J^2/k-Q_{I\bs J}^2/(n-k)]\ge 0$, then
\be G(I,J,t)\le e^{-\cq t[Q_J^2/k+Q_{I\bs J}^2/(n-k)]}\int_0^t
{ds\over \sqrt{t-s}} \le 2e^{-\cn tQ_I^2} \lb{a19} \ee
where we used the fact that $t\le 1$ and the inequality, valid for two
arbitrary real numbers $a$ and $b$:
\be {a^2\over k} + {b^2\over n-k} \ge {1\over n} (a+b)^2 \ee
If $[Q_I^2-Q_J^2/k-Q_{I\bs J}^2/(n-k)]\le 0$, then
\be G(I,J,t)\le e^{-\cq tQ_I^2} \int_0^t
{ds\over \sqrt{t-s}} \le 2e^{-\cn tQ_I^2} \lb{a21} \ee
\pref{a17}, \pref{a19} and \pref{a21} imply that:
\be E(I,t) \le 2^{n-2}e^{-\cn Q_I^2t} \sum_{k=1}^{n-1}
{n\choose k}k^{k-1}(n-k)^{n-k-1} \le (2n)^{n-1}e^{-\cn Q_I^2t} \ee
where we used the identity (see \cite{BK}: Lemma 4.2):
\be \sum_{k=1}^{n-1}
{n\choose k}k^{k-1}(n-k)^{n-k-1} = 2(n-1)n^{n-2} \ee

Then \pref{a16} is proved; if we insert it into \pref{a14}, we obtain the bound
\pref{a11}.
%
%\bibliography{coul_gas}
%
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\end{document}
ENDBODY
