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\begin{document}
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\title{Charge correlations for the two dimensional Coulomb gas\thanks{%
Talk presented at the Conference on Constructive Results on Field Theory,
Statistical Mechanics and Condensed Matter Physics, Palaiseau, July 25-27,
1994.}}
\author{T. R. Hurd\thanks{Research
supported by the Natural Sciences and Engineering Research Council of
Canada.}\\Department of Mathematics and Statistics\\McMaster
University\\Hamilton,
Ontario\\L8S 4K1}
\maketitle
\babs This paper is a summary of mathematical results contained in
\cite{Hur94c} concerning integer charge correlations for the Coulomb
gas/sine-Gordon system in two dimensions. For $\beta=T^{-1}<8\pi$ and small
activity $z$, the UV problem is considered in a finite volume. A new proof is
given of the fact that the pressure $p^{>m}(\beta,z)$, renormalized up to order
$m$ in perturbation theory, is analytic in $z$ for $\beta<\beta_m=8\pi(1-1/m)$.
Higher correlations are treated and proven to be analytic in $z$ for all
$\beta<8\pi$. The $m$th threshold value $\beta_m$ appears as the value at which
the exponent of the short distance power law of the $m$th subleading
contribution
to any correlation changes nonanalytically. In the Kosterlitz-Thouless phase
$\beta>8\pi$, the IR problem is treated with a fixed UV cutoff. The existing
framework for the pressure is extended to all higher correlations. For the two
point function, it is shown that at length scales larger than
$\cO(|z|^{-1/(\beta/4\pi-2)})$ the free field power law
$|x-y|^{-\beta/2\pi}$ at long distances crosses over to a slower power law
$|x-y|^{-4}$. This verifies a conjecture of Fr\"{o}hlich and Spencer
\cite{FrSp80}. \eabs
\section{Introduction}
The two dimensional Coulomb
gas/sine-Gordon system is a classic model in mathematical physics. There
exists a
large body of discussion of its special properties, including landmark
papers such as that of Kosterlitz and Thouless \cite{KoTh73}, Coleman
\cite{Col75}, Zamolodchikov \cite{Zam79}, and Sklyanin et al.
\cite{SkTaFa80}. On
the level of constructive quantum field theory, there is a somewhat smaller body
of results: these include the works of \cite{FrSe76}, \cite{FrSp80},
\cite{FrSp81}, \cite{BrFe80},\cite{BeGaNi82},\cite{MaKlPe90}.
My aim here is
to extend the constructive renormalization group (RG) programme developed in a
series of
papers
\cite{DiHu91},\cite{DiHu92a},\cite{DiHu92b},\cite{DiHu93},\cite{BrDiHu94a}
which began with a foundational paper \cite{BrYa90} by Brydges and Yau. I
shall
explain how the BY method,
with some adaptations based on ideas in \cite{BrKe94},
leads to a rather complete picture of
perhaps the most important aspect of the model, namely the asymptotics of
integer
charge correlations, both in the ultraviolet (UV) and infrared (IR).
We consider the Coulomb gas with
activity $z$ at temperature $T=\beta^{-1}$ on a finite torus $\La(M)={{\bf
R}^2}/{(L^M{\bf Z})^2}$ where $L>1, M$ are integers. Let the Coulomb potential
be the inverse Laplacian on $\La(M)$ with a short
distance cutoff at scales $\sim L^{-N}$:
\be\label{VMN} v_{M,N}(x)={L^{-2M}}\sum_{{p\in\La(M)^*}\atop{p\ne
0}}p^{-2}{e^{ipx}\ e^{-L^{-2N}p^2}},\ee
where $\La(M)^*$ denotes the dual lattice $(2\pi L^{-M}{\bf Z})^2$. Then
the system
is described by the grand canonical partition function
\[Z_{CG}(\beta,z,v_{M,N})=\sum_{n,{\vec q}}
\int_{\Lambda(M)^n} d{\vec x}\ {z^n\over n!}\ \exp (-\beta
E_{n,v_{M,N}}(\vec x;\vec
q) )\]
where the Coulomb energy is given by
\[E_{n,v_{M,N}}(\vec x;\vec q)=\sum _{a**0$, we see that the SG model
naturally leads to a suppression of non-neutral configurations. It appears, as
indeed we shall show, that the {\it non-Wick-ordered}
sine-Gordon model is an appropriate starting point for the IR problem. The
difference between (\ref{zsgw}) and (\ref{zsg}) is a divergent factor
\be\label{tau}\tau_{M,N}=e^{\beta v_{M,N}(0)/2}\sim L^{\beta(M+N)/4\pi}\ee
multiplying the coupling constant $\z$.
Now consider a configuration $\Si=\{\xi_1,\dots,\xi_Q\}$ of $Q$ external unit
charges $\xi_a=(x_a,q_a)$, $q_a=\pm 1$, fixed at points $x_a$ in
$\La(M)$. To this we associate a partition function
$Z^\Si$ where the Coulomb energies coming from the external charges are
included.
The {\it integer charge truncated correlations} are defined to be:
\[\left\{\barr{cl}
S^{q_1}_{x_1}=Z^{-1}
Z^{q_1}_{x_1}=(2z)^{-1}\times{\rm density }&Q=1\\ &\\
S^{q_1q_2}_{x_1x_2}=Z^{-1}
Z^{q_1q_2}_{x_1x_2}-Z^{-2} Z^{q_1}_{x_1}Z^{q_2}_{x_2}& Q=2\\& \\
{\rm etc.}& Q>2\earr\right.\]
Now, for the UV problem, we define a generating function \be\label{gfw}
Z(\vec\lambda)=\int d\mu_{\beta
v_{M,N}}(\phi)\prod^Q_{a=1}(1+\lambda_a:e^{iq_a\phi(x_a)}:)
e^{\zeta\int:\cos\phi:}
\ee
For the IR problem, we define
\be\label{gf}
Z(\vec\lambda)=\int d\mu_{\beta
v_{M,N}}(\phi)\prod^Q_{a=1}(1+\lambda_a e^{iq_a\phi(x_a)}) e^{\zeta\int\cos\phi}
\ee
Then, in both cases, the integer
charge truncated correlations are given by differentiating with respect to
$\vec\la$: \[\left\{\barr{cl}
S^{q_1}_{x_1}=\left(\frac{\pa}{\pa\la_1}\right)\log
Z(\vec\la)\Big|_{\vec\la=0}&Q=1\\&\\
S^{q_1q_2}_{x_1x_2}=\left(\frac{\pa^2}{\pa\la_1\pa\la_2}\right)\log
Z(\vec\la)\Big|_{\vec\la=0}& Q=2\\&\\ {\rm etc.}& Q>2\earr.\right.\]
Fr\"{o}hlich and Seiler (\cite{FrSe76}, p 899), have observed that for the
sine-Gordon quantum field theory, ``... these fields (i.e. $:e^{iq\f}:$) are
actually much more natural than the field $\f$.'' Field correlations
such as $\left<\f(x)\f(y)\right>$ have been treated in the sine-Gordon model
in \cite{DiHu93}.
In this review, I will describe how the recent refinements of the
renormalization group (RG) in constructive quantum field theory can be
extended to
give a rather complete picture of integer charge correlations.
Sections 2 and 3 will describe how the main theorems described in \S 5,6
(Theorems
\ref{T2} and \ref{TIR2}) lead to
detailed results on the behavior of correlations, for the UV problem
with $\beta<8\pi$ and the IR problem for $\beta>8\pi$, respectively. I will
discuss how these new results relate to and improve on previously existing
results.
In section 4, I will sketch the Brydges and Yau RG method and show how it
extends
to correlation functions. Then in sections 5 and 6, I will state Theorems
\ref{T2}
and \ref{TIR2}. These are the main technical results dealing with the UV
problem and IR problem respectively. In \S6, I provide a
thumbnail sketch of the proof of the vacuum IR result (Theorem
\ref{TIR}) to give a flavour of how these things go. For the complete proofs
of the results on correlations, the reader is directed to the original paper
\cite{Hur94c}.
\section{Pressure and correlations for $\beta<8\pi$} In this
temperature range, we concentrate on the UV limit $N\rightarrow\infty$ in a
fixed
volume $\La(0)$ (equally we could work in infinite volume by introducing a
mass term in the Gaussian measure). We fix the RG rescaling parameter $L$ to be
a large integer, take a parameter $0<\ep<1/2$, and take a complex
activity parameter $\z=2z$ which is ``small'', i.e.
$|\z|\le\bar\z(\beta,\ep,L)$.
The simplest consequence of the RG analysis of \S 5 is a formula
(\ref{press}) for
the pressure as a sum over length scales $L^{-i}$, $i=0,1,\dots, N$. In the
$N\rightarrow \infty$ limit, this has the form
\[p(\z)=\sum_{i=0}^\infty
\Omega_i(\z).\]
>From Theorem \ref{T2}, we find that each quantity $\Omega_i$ is analytic in
$\z_i=L^{-2i}\tau_{i,0}\z\sim L^{(\beta/4\pi-2)i}\z$ with radius $\bar \z$,
and is
bounded there by \[L^{2i}|\z_i|^{2-\ep}.\] This result is somewhat
stronger than
the similar
inequality (35) of \cite{DiHu93}.
Because of analyticity, we can immediately derive bounds on the power series
about $\z=0$. Write
\[p(\z)=\sum_{i=0}^{m-1} \z^i p^{(i)} + p^{\ge m}\]
where
\beaa \z^mp^{(m)}&=&\frac{1}{2\pi i}\oint\frac{ds}{s^{m+1}}p(s\z)\\
p^{\ge m}&=&\frac{1}{2\pi i}\oint\frac{ds}{s^{m}(s-1)}p(s\z)
\eeaa
and similar expressions for $\Omega^{(m)},\Omega^{\ge m}$. For each
$\Omega_i$, we
can bound the integral over the contour $|s|=(L^{-2i}\tau_{i,0}|\z|)^{-1}\bar
\z\gg 1$, and conclude \bthm \benum \item For any even integer $m\ge 2$,
the $m$th
order contribution to the pressure is bounded by
\be\label{pbound}|p^{(m)}|\le{\bar\z}^{2-m-\ep}\ \sum_{i=0}^\infty\
L^{[2-m(2-\beta/4\pi)]i}\ee
which converges for $\beta<\beta_m=8\pi(1-1/m)$.
\item The $m$th renormalized pressure $p^{\ge m}=\sum_{a=m}^\infty p^{(m)}$
is bounded by
\[|p^{\ge m}|\le\ 2\ |\z|^m\ {\bar\z}^{2-m-\ep}\ \sum_{i=0}^\infty\
L^{[2-m(2-\beta/4\pi)]i}\]
which converges for $\beta<\beta_m$.
\eenum\ethm
Thus for $\beta\in[\beta_m,\beta_{m+2})$, the interaction
needs only a vacuum renormalization up to counterterms of order $m$. This
result is
an independent and very much simplified proof of what is essentially
Theorem 1.0 of
\cite{NiReSt86}.
A similarly comprehensive picture holds for the general truncated integer charge
correlations. Let $\Si$ be a configuration of $Q>1$ external charges where the
interparticle separations are all of order $L^{-I}$ for some integer $I>0$, i.e.
\[L^{-(I+1)}m}|\le C_2^{Q}Q!\left(\frac{|\z|}{\bar\z}\right)^m
d(\Si)^{-\De(\Si)}\] where $d(\Si)=L^{-I}$ and the exponent of the short
distance
power law is \be\label{powers}\left\{\barr{cl}
{(Q-1)\beta\over
4\pi}+\epsilon Q(2-{\beta\over
4\pi})
& \mbox {for $\beta\le\beta_{m+1}$}\\&\\
{Q\beta\over
4\pi}-m(2-{\beta\over
4\pi})+\epsilon Q(2-{\beta\over
4\pi})
& \mbox {for
$\beta>\beta_{m+1}$}\earr\right.
\ee
\ethm
This result is new. In fact, I have not been able to find any perturbative or
constructive statement in the literature on the short distance asymptotics of
general correlations.
A direct comparison with
perturbative bounds obtained by extending the tree expansion technique of
\cite{BeGaNi82} suggests that the above exponents for each $S^{(m)}$ are
in fact
sharp if we take $\ep=0$.
\bigskip \nind {\bf Example:}
Two-point function,
$\beta\in(\beta_8,\beta_9]=(7\pi,72\pi/9]$:
$$\cases{
S^{(0)}\sim
d(\Sigma)^{-[2\beta/4\pi]}& \cr\cr
S^{(2)}\sim
d(\Sigma)^{-[2\beta/4\pi-2(2-\beta/4\pi)]}& \cr\cr
S^{(4)}\sim
d(\Sigma)^{-[2\beta/4\pi-4(2-\beta/4\pi)]}& \cr\cr
S^{(6)}\sim
d(\Sigma)^{-[2\beta/4\pi-6(2-\beta/4\pi)]}& \cr\cr
S^{(\ge 8)}\sim
d(\Sigma)^{-[\beta/4\pi+2\epsilon(2-\beta/4\pi)]}&
\cr\cr}$$
In general, for $\beta\in(\beta_m,\beta_{m+1}]$ we shall see a finite linear
sequence of exponents, which at the $m$th level stabilize at the value
${(Q-1)\beta\over 4\pi}$. We can see that the thresholds
$\beta_m$ are values at which the exponent of the short distance power law
of the
$m$th order two-point function changes non-smoothly. This crossover
phenomenon can
be expected on the basis of a perturbative analysis.
Note that correlations remain smooth even at the thresholds.
\section{Results for $\beta>8\pi$}
In this low temperature range, we consider the IR problem, with a fixed UV
cutoff
at the unit distance scale. The volume is taken to be $\La(M)$, where $M$ is
arbitrarily large. The main theorem for this problem is stated in \S 6: Here
I describe the picture of the pressure and correlations which follows from it.
Again, the parameter $\z$ is chosen ``small'', i.e.
$|\z|\le\bar\z(\beta,\ep,L)$.
\bigskip
\nind{\bf Apology:} In order to be consistent with the UV picture, scale labels
$i,j$ are negative in the IR regime and correspond to large length scales
$L^{|i|}$.
\bthm The finite volume pressure is given by a sum
over scales: \[p(\La(M))=\sum_{j=0}^M \Omega_{-j}.\]
Each term
$\Omega_{-j}(\z)$ is analytic for $|\z|<\bar\z$, and bounded there by
$L^{-2|j|}D\de^{|j|}$, where $\de<1/2$ and $D$ is a small constant.
Therefore the
pressure
itself is analytic in $\z$ and since it vanishes to first order in $\z$,
is bounded by $\cO(|\z|^2)$, uniformly in $M$. \ethm
Now consider the general truncated integer charge correlations. As in \S 2, I
suppose that the configuration $\Si$ is such that the interparticle separations
are all of the same order $L^{|I|}$ for some negative integer $I$. The theorem
implies that there is a crossover size
\[L^{|I_0|}\sim|\z|^{\frac{-1}{\beta/4\pi-2}}
\]
where the power law decay
rate shifts. At distances shorter than this scale, the power is that of the free
field
$|x-y|^{-\beta/2\pi}$; a larger scales, the decay is the faster
$|x-y|^{-4}$.
The RG
leads to an expansion \[S^\Si=\sum_{j=|I|}^M D_\Si\Omega_{-j}. \]
Theorem \ref{TIR} implies that each $D_\Si\Omega_{-j}$ is analytic in
$\z$ with radius $\bar \z$, and bounded by
\[\left\{\barr{lr}
Q!(c' L^{-2})^{|i-I|} \left(c_1L^{-\beta |I|/4\pi}\
\right)^Q&\mbox{if $|I|\le|I_0|$}\\
Q!(c' L^{-2})^{|i-I|} \left(c_2|\z|L^{-2|I|}\
\right)^Q &\mbox{if $|I|\ge|I_0|$}
\earr\right.
\]
for some quantities $c_i=c_i(\beta,\ep,L)$ and
$c'=\cO(1)$.
Thus in all cases, the sum over $j$ converges uniformly in $M$.
\bthm The truncated correlation of a configuration $\Si$ of $Q>1$ points with
separation $d(\Si)=L^{|I|}$ is bounded above by
\[c^Q Q! d(\Si)^{-\De(\Si)}\]
where
\[\De(\Si)=
\left\{\barr{lr}
\frac{\beta}{4\pi}Q&\mbox{if $|I|\le|I_0|$}\\
2Q&\mbox{if $|I|\ge|I_0|$}
\earr\right.
\]
The value of $c$ is
\[\left\{\barr{lr}
c_1&\mbox{if $|I|\le|I_0|$}\\
c_2|\z|&\mbox{if $|I|\ge|I_0|$}
\earr\right.
\]\ethm
This result verifies a conjecture which goes back
to \cite{FrSp80}. They state that the two-point correlator should decay at large
distances like the dipole-dipole correlation in a dipole gas, averaged over the
dipole direction (i.e. like $|x|^{-4}$).
If we combine these upper bounds with an analysis of low order perturbation
theory, we presumably obtain the same power
laws as lower bounds on correlations. For example, an explicit calculation
should show that the second order contribution to the two point correlator
has the
exact power law $|x|^{-4}$, and that for small $\z$ the higher order
contributions
are negligible.
\section{RG maps for charge correlations}
The remainder of the paper will sketch the method by which the results of \S2,3
are proved. We consider in more detail the UV problem $N\rightarrow\infty$ in a
finite volume $\La(0)$, for $\beta<8\pi$. We take a configuration
$\Si=\{\xi_1,\dots,\xi_Q\}$ of $Q$ external unit charges $\xi_a=(x_a,q_a)$,
$q_a=\pm 1$,
located at points $x_a$ in $\La(0)$. The truncated $Q$-point
correlation function is defined to be:
\[S^\Si=\frac{\pa^Q}{\pa\la_1\dots\pa\la_Q}|_{\vec\la= 0}\log Z(\vec\la),\quad
\vec\la=(\la_1,\dots,\la_Q),\] where
\be\label{zlamb} Z(\vec\la)=\int d\mu_{\beta
v_{N,0}}(\f)\prod^Q_{a=1}(1+\la_a\tau_{N,0} e^{iq_a\phi(L^Nx_a)})\cZ^N_N(\f).\ee
The
functional integral has been rewritten on the rescaled volume
$\La(N)=L^N\La(0)$.
The ``Boltzmannian'' or ``Gibb's Factor'' $\cZ^N_N$ is given by
\[\cZ^N_N(\f)=\exp[\z_N\int_{\La(N)}\cos\f],\]
where $\z_N=L^{-2N}\tau_{N,0}\z$
with
\[\tau_{N,0}=e^{\beta v_{N,0}(0)/2}\sim L^{\beta N/4\pi}\]
being just the
Wick-ordering constant. The truncated 2-point function, for example, is
\[S^{q_1q_2}_{x_1x_2}=\tau_{N,0}^2\left
_{\beta,N,\z}-
\tau_{N,0}^2\left_{\beta,N,\z}\left
_{\beta,N,\z}\]
We define the following operators
on linear functions of $\vec\la$: for each
$\si\subset\{1,\dots,Q\}$, let \bea
\la_\si &=&\prod_{a\in\si}\la_a\\
D_\si\ \cdot&=&\prod_{a\in\si}\frac{\pa\ \cdot}{\pa\la_a}\Big|_{\vec\la=0}\\
P_\si&=&\la_\si\ D_\si
\eea
The identity operator on linear functions of $\vec\la$ can be written
\[P=\sum_\si P_\si=P_\emptyset+P_>.\]
For any $\Si$, the
correlation function $S^\Si$ is given by
\[ D_\Si \log Z.\]
Our analysis is thus concerned with generating a convergent formula for
$Z(\vec\la)$.
Now I shall give a brief sketch of the RG maps developed for the partition
function in the sequence of papers
\cite{BrYa90},\cite{DiHu91},\cite{DiHu93}. The
reader should ideally refer to the last of these papers for a complete
description.
The BY method generates a sequence of measures $d\nu_i$ on certain
function spaces over the volumes $\La(i)$, for $i=0,1,\dots, N$. The measures of
interest are always weak perturbations
\[d\nu_i(\f)=d\mu_{\beta v_{i,0}}(\f)\ \cZ_i(\La(i),\f)\]
of a specific sequence of Gaussian measures $d\mu_{\beta v_{i,0}}$ whose
covariances $\beta v_{i,0}(x,y)$ are given by (\ref{VMN}). The
Gibb's factor is taken in the form of a polymer expansion:
\beaa\cZ^N_i(\La(i),\f)&=&\sum_{X_1,\dots,X_L}\prod_j K_i(X_j,\f)\\ &\equiv&\cE
xp[\Box+K_i](\La(i),\f)\eeaa where the sum is over disjoint collections of
``polymers'', a polymer $X$ being a union of closed unit squares with
corners lying
on the integer lattice in $\La(i)$. The ``$\cE xp$'' notation may be
thought of as
shorthand for the polymer expansion. The collection of polymer activities $
K=\{K_i(X,\f), X\mbox{ a polymer}\}$ is called an {\it analytic functional}.
The analytic functionals are to be regarded as lying in
Banach spaces $\cB_i$, whose norms $\|\cdot\|_i$ are given
weights parametrized by certain quantities:
\benum\item the large field parameter $\k_i>0$;
\item the large set parameter $A>0$;
\item the large derivative parameters $\bh_i=(h_{0i},h_{1i})$.
\eenum
Now the RG map $\cR_i$ is a specific nonlinear functional which takes a ball in
$\cB_i$ into $\cB_{i-1}$. $\cR$ is quite naturally the
composition of three maps $\cS\circ\cE\circ\cF$, called scaling,
extraction and fluctuation. Taken all together, the map $\cR$ has
the following defining property. Let the {\it
fluctuation covariance} be defined by the Fourier components
$c_i(p)=v_{i,0}(p)-L^2v_{i-1,0}(Lp)$ for $i>1$ and $c_1(p)=v_{1,0}(p)$ for
$i=1$.
Then for any $K_i\in\cB_i$ small enough, $K_{i-1}=\cR(K_i)$ is such that \[\cE
xp[\Box+K_{i-1}](\La(i-1),\f)=e^{\Omega_i} \mu_{\beta c_i}*\cE
xp[\Box+K_{i}](\La(i),\f_L) \] Here, $\mu*$ denotes Gaussian convolution,
$\Omega_i$ is a certain carefully chosen constant, and $\f_L$ denotes the
rescaled
field $\f_L(x)=\f(L^{-1}x)$. This defining property implies in particular that
\[\int d\mu_{\beta v_{i,0}}\cZ_i=e^{\Omega_i}\int d\mu_{\beta
v_{i-1,0}}\cZ_{i-1}\]
The central idea of the RG is that by iteration starting from the measure with
\[\cZ^N_N(\f)=\exp \z_N\int_{\La(N)}\cos\f=\cE xp[\Box+K^N_N](\f),\]
the
UV regularized partition function can be written
\[Z_{SG}(\zeta,\beta,v_{0,N})=\left(\prod_{j>i}^Ne^{\Omega^N_j}\right)\int
d\mu_{\beta v_{i,0}}\cZ_i^N\] for any integer $i=1,2,\dots,N$, or for $i=0$
\[Z_{SG}(\zeta,\beta,v_{0,N})=\left(\prod_{j>0}^Ne^{\Omega^N_j}\right)
\cZ_0^N(\La(0),\f=0).\]
The regularized pressure is
\be\label{press}
p(\zeta,\beta,v_{0,N})=\sum_{j=1}^N \Omega_j^N+\log \cZ^N_0.\ee
Now the UV problem can be solved by controlling the limit
\[\cZ_i=\lim_{N\rightarrow\infty}\cZ_i^N\]
Now, following a similar idea introduced in \cite{BrKe94}, we can develop the
analogous treatment for the $\la$-dependent partition function, by extending the
activities $K^N_N$ to $\la$-dependent activities so that
\be\label{gibbsN}
\cE
xp[\Box+K^N_N(\vec\la)](\La(N))=\prod^Q_{a=1}(1+\la_a\tau_{N,0}e^{iq_a\f(L^N
x_a)})\cZ^N_N(\f).\ee
There is an essential distinction between vacuum and non-vacuum
activities: $P_\emptyset K^N_N(X)$ is translation invariant, whereas $P_\Si
K^N_N(X)$ is zero unless $X$ contains all of the points in the set
$L^N\Si$. This
``pinning'' is preserved under the RG and leads at each scale to a good
powercounting factor of $L^{-2}$ for non-vacuum activities.
We define a new
renormalization group map
\[\cR_P=\cS\circ\cE_P\circ\cF:K_i\goesto
K_{i-1}=\cR_P(K_i)\]
such that
\[
\int d\mu_{v_{i,0}}\ \cE
xp(\Box+K_i(\vec\la))(\Lambda(i))=P\left(e^{\Omega_i}\ \int d\mu_{v_{i-1,0}}\
\cE xp(\Box+K_{i-1}(\vec\la)\right)(\La(i-1)).\]
It turns out that the only modifications needed to define $\cR_P$ are to the
extraction step $\cE_P$, and lead to no new difficulties.
Applying this iteration $N$ times to the formula (\ref{zlamb}) leads to an
expansion for $Z(\vec\la)$: \[ Z(\vec\la)=P\left(\left(\prod_{i=1}^N
e^{\Omega_i^N}\right)\cE xp(\Box+K^N_0)(\La(0))\right)\]
Since $K^N_0$ is defined on a unit block,
\[\cE xp(\Box+K^N_0)(\La(0))=1+P_\emptyset K^N_0(\De)+P_>
K^N_0(\De,\vec\la)=P e^{\Omega_0^N(\vec\la)}\]
where
\[\Omega_0^N(\vec\la)=P\log (1+P_\emptyset K^N_0(\De)+P_> K^N_0(\De,\vec\la)).\]
Then
\[Z(\vec\la)=P\left(\prod_{i=0}^N
e^{\Omega_i^N}\right)\]
which gives the desired formula for any truncated correlation:
\bea\label{expansion}S^\Si=\sum_{i=0}^N \left(D_\Si \Omega^N_i\right).\eea
\section{The main result for $\beta<8\pi$}
We note a simplifying property of the
expansion (\ref{expansion}): since extractions are only made from ``small''
sets,
$D_\Si \Omega^N_i=0$ if $L^i\Si$ is not contained in some small set. This
means that for two or more distinct points $\Si$, the
expansion (\ref{expansion}) has a finite number of terms which depends on the
geometry of the points.
For simplicity, we make an assumption on the configuration $\Si$ that the
interparticle distances are all of the order $L^{-I}$ for some integer $I$:
\[L^{-(I+1)}1$
\[\|D_\Si K_i\|_{i-j}\le \left\{\barr{lr}
Q! \left(2AL^{\beta i/4\pi}\
e^{h_{0i-j}}\right)^Q&\mbox{if $i\ge I$}\\
Q!(c_1 L^{(\beta/4\pi)})^{(i-I)} \left(2AL^{\beta I/4\pi}\
e^{h_{0I-j}}\right)^Q &\mbox{if $i\le I$}
\earr\right.\]
and
\be|D_\Si\Omega^N_i|\le\left\{\barr{lr}
0 &\mbox{if $i > I$}\label{igeI}\\
Q!(c_1 L^{(\beta/4\pi)})^{(i-I)} \left(2AL^{\beta I/4\pi}\
e^{h_{0I}}\right)^Q &\mbox{if $i\le I$}\label{ileI}
\earr\right.\ee
\eenum
\ethm
\proof \ \ The proof, given in \cite{Hur94c}, follows with moderate
changes the
proof of the vacuum result in \cite{DiHu93}. \QED
\bigskip
This result leads rather quickly to the claimed bounds of \S 2.
\section{Sketch of the method and results for $\beta>8\pi$}
For the Kosterlitz-Thouless phase $\beta>8\pi$, one takes the
initial measure $d\nu_0$ to be a weak perturbation of the Gaussian measure on a
large volume $\La(M)$ with covariance $v_{M,0}$ (with short distance cutoff
$N=0$).
The RG map is defined in a similar way to the map defined for $\beta<8\pi$. For
each negative integer $i=0,-1,\dots,-(M-1)$, it takes activities $K_i$ on volume
$\La(M+i)$ to activities on volume $\La(M+i-1)$. There is one important
difference
in the definition of $\cR$: the extraction step removes a factor
$e^{\de\si_i\int_{\La(M+i)}(d\f)^2}$ from $\cE xp [\Box + K_i]$, in addition to
the vacuum extraction $\Omega_i$. These wave function renormalizations are
accumulated in the covariances \[\hat v_i(p)^{-1}=p^2(e^{p^2}+\si_i)\] where
$\si_i=\sum_{j=1}^{|i|}\de\si_{-j}$. Changing the normalization
of the Gaussian measure leads to an additional contribution $\Omega'$ to the
vacuum term
$\Omega_i$. To adequately control the extra renormalization cancellations, it is
now necessary to consider functionals of $(\f,\pa_i\f,\pa_i\pa_j\f)$, and the
corresponding derivative regulators $\bh=(h_0,h_1,h_2)$.
The following parameters lead to a contractive estimate for $\cR$. Here,
$c_2,c_3,c_4,c_5$ are certain $\cO(1)$ geometric constants. We fix
$\ep>0$ and a large integer $L$ so that
\[\de\equiv c_2\max(L^{2-(1-\ep)\beta/4\pi},L^{-2})<1/2.\]
We take the regulators as follows:
\benum
\item $ G_i$ as before, with \[\k_i=c_3(1+\sum_{j=1}^{|i|}\de^j);\]
\item $\G$ as before, with constant $A=\cO(L^{2+\ep});$
\item $\bh_i=h_i(c_4 L,L,L^2)$ where
$h_i=c_5\beta(2-\sum_{j=1}^{|i|}\de^j)$. \eenum
Note that $\frac{1}{2}h_08\pi$
it is the wave function
terms which dominate the asymptotics. However, the picture is complicated,
because
wave function terms enter only in second order perturbation theory. This means
that for small coupling constant there is a crossover at a scale $I\sim I_0$
with
\[I_0\sim\frac{1}{(\log L)(2-\beta/4\pi)}\log(L^2 e^{h_{00}}|\z|)\]
For $|I|\le |I_0|$ one sees the power law $|x-y|^{-\beta/2\pi}$ induced by the
charge $1$ terms, while for $|I|>|I_0|$ one sees the $|x-y|^{-4}$ law of
the wave
function terms.
All of this is a consequence of the following theorem on general correlations,
whose proof, given in \cite{Hur94c}, amounts to a moderate extension of the
above vacuum result.
\bthm\label{TIR2} There is a number $\bar\z(\ep,L,\beta)$ and a geometric
constant $c_1$ such that the following properties hold,
uniformly in the volume $\La(M)$, and hence in the thermodynamic limit for all
$i=0,-1,\dots,-M$: \benum
\item (analyticity) The activities $D_\Si K_i$ and extracted parts
$D_\Si\Omega_i$ are analytic functions of $\z$ for
$|\z|\le\bar\z$.
\item (one point function) For any
$\Si=\{(x,q)\}$,
\bea\label{oneptc}
\|D_\Si K_i\|_{i} &\le& \left\{\barr{lr}
2A L^{-\beta |i|/4\pi}\
e^{h_{0i}}&\mbox{if $|i|\le |I_0|$}\\
2AL^2 L^{-2|i|}\
e^{2h_{0i}}|\z|&\mbox{if $|i|\ge |I_0|$}
\earr\right.\\
\label{oneptd}
|D_\Si\Omega^N_{i}|&\le&\left\{\barr{lr}
2A L^{-\beta |i|/4\pi}\
e^{h_{0i}}&\mbox{if $|i|\le |I_0|$}\\
2AL^2 L^{-2|i|}\
e^{2h_{0i}}|\z|&\mbox{if $|i|\ge |I_0|$}
\earr\right.
\eea
\item (multi-point functions) Let $\Si$ be a configuration of $Q>1$ points
with linear separation of order $L^{|I|}, I<0$.
\benum
\item Suppose $|I|\le |I_0|$. Then
\[\|D_\Si K_i\|_{i}\le \left\{\barr{lr}
Q! \left(2AL^{-\beta |i|/4\pi}\
e^{h_{0i}}\right)^Q&\mbox{if $|i|<|I|$}\\
Q!(c_1 L^{-2})^{|i-I|} \left(2AL^{-\beta |I|/4\pi}\
e^{h_{0I}}\right)^Q &\mbox{if $|i|\ge |I|$}
\earr\right.\]
and
\be|D_\Si\Omega^N_i|\le\left\{\barr{lr}
0 &\mbox{if $|i|<|I|$}\label{igeI2}\\
Q!(c_1 L^{-2})^{|i-I|} \left(2AL^{-\beta |I|/4\pi}\
e^{h_{0I}}\right)^Q &\mbox{if $|i|\ge |I|$}\label{ileI2}
\earr\right.\ee
\item Suppose $|I|\ge |I_0|$.
\[\|D_\Si K_i\|_{i}\le \left\{\barr{lr}
Q! \left(2AL^{-\beta |i|/4\pi}\
e^{h_{0i}}\right)^Q&\mbox{if $|i|< |I_0|$}\\Q! \left(2AL^2 L^{-2|i|}\
e^{2h_{0i}}|\z|\right)^Q&\mbox{if $|I_0|\le |i|\le |I|$}\\
Q!(c_1 L^{-2})^{|i-I|} \left(2AL^{2}L^{-2|I|}\
e^{2h_{0I}}|\z|\right)^Q &\mbox{if $|i|\ge |I|$}
\earr\right.\]
and
\be|D_\Si\Omega^N_i|\le\left\{\barr{lr}
0 &\mbox{if $|i| < |I|$}\label{igeI3}\\
Q!(c_1 L^{-2})^{|i-I|} \left(2AL^2 L^{-2|I|}\
e^{2h_{0I}}|\z|\right)^Q &\mbox{if $|i|\ge |I|$}\label{ileI3}
\earr\right.\ee
\eenum
\eenum
\ethm
\section{Acknowledgements} I am indebted to G. Benfatto, whose insight and key
observations precipitated the present work, and to G. Keller and D. Brydges for
telling me at an early stage about their approach to correlations. I would also
like to thank V. Rivasseau and K. Gawedzki for their interest, comments and
support, and to acknowledge the support of the following institutions where some
of this work was done: Institut
des Hautes Etudes Scientifiques, France; Ecole Polytechnique, France; and
Universit\`{a} di Roma II.
\begin{thebibliography}{10}
\bibitem{BeGaNi82}
G.~Benfatto, G.~Gallavotti, and F.~Nicol\`{o}.
\newblock On the massive sine-{G}ordon equation in the first few regions of
collapse.
\newblock {\em Commun. Math. Phys.}, 83:387--410, 1982.
\bibitem{BrDiHu94a}
D.~Brydges, J.~Dimock, and T.R. Hurd.
\newblock The short distance behavior of $\phi^4_3$.
\newblock McMaster University preprint, 1994.
\bibitem{BrFe80}
D.~Brydges and P.~Federbush.
\newblock Debye screening.
\newblock {\em Commun. Math. Phys.}, 73:197--246, 1980.
\bibitem{BrKe94}
D.~Brydges and G~Keller.
\newblock Correlation functions of general observables in dipole-type systems
{I}: accurate upper bounds.
\newblock University of Virginia preprint, 1994.
\bibitem{BrYa90}
D.~Brydges and H.~T. Yau.
\newblock Grad $\varphi$ perturbations of massless {G}aussian fields.
\newblock {\em Commun. Math. Phys.}, 129:351--392, 1990.
\bibitem{Col75}
S.~Coleman.
\newblock Quantum sine-{{G}ordon} equation as the massive {T}hirring model.
\newblock {\em Phys. Rev.}, D11:2088--2097, 1975.
\bibitem{DiHu91}
J.~Dimock and T.~R. Hurd.
\newblock A renormalization group analysis of the {Kosterlitz-{T}houless}
phase.
\newblock {\em Commun. Math. Phys.}, 137:263--287, 1991.
\bibitem{DiHu92b}
J.~Dimock and T.~R. Hurd.
\newblock A renormalization group analysis of correlation functions for the
dipole gas.
\newblock {\em J. Stat. Phys.}, 66:1277--1318, 1992.
\bibitem{DiHu92a}
J.~Dimock and T.~R. Hurd.
\newblock A renormalization group analysis of {QED}.
\newblock {\em J. Math. Phys.}, 33:814--821, 1992.
\bibitem{DiHu93}
J.~Dimock and T.~R. Hurd.
\newblock Construction of the two-dimensional sine-{G}ordon model for
$\beta<8\pi$.
\newblock {\em Commun. Math. Phys.}, 156:547--580, 1993.
\bibitem{FrSe76}
J.~Fr{\"o}hlich and E.~Seiler.
\newblock The massive {Thirring-Schwinger} model ${QED}_2$. {C}onvergence of
perturbation theory and particle structure.
\newblock {\em Helv. Phys. Acta.}, 49:889--924, 1976.
\bibitem{FrSp80}
J.~Fr{\"o}hlich and T.~Spencer.
\newblock On the statistical mechanics of classical {C}oulomb and dipole gases.
\newblock {\em J. Stat. Phys.}, 24:617--701, 1980.
\bibitem{FrSp81}
J.~Fr{\"o}hlich and T.~Spencer.
\newblock The {Kosterlitz-Thouless} transition in two-dimensional {A}belian
spin systems and the {C}oulomb gas.
\newblock {\em Commun. Math. Phys.}, 81:527--602, 1981.
\bibitem{Hur94c}
T.~R.~Hurd.
\newblock Charge correlations for the {C}oulomb gas.
\newblock McMaster University preprint, 1994.
\bibitem{KoTh73}
J.~M. Kosterlitz and D.~J. Thouless.
\newblock Ordering, metastability and phase transitions in two-dimensional
systems.
\newblock {\em J. Phys.}, C6:1181--1203, 1973.
\bibitem{MaKlPe90}
D.~H.~U. Marchetti, A.~Klein, and J.~F. Perez.
\newblock Power law fall off in the {Kosterlitz-Thouless} phase of a
two-dimensional lattice {C}oulomb gas.
\newblock {\em J. Stat. Phys.}, 60:137, 1990.
\bibitem{NiReSt86}
F.~Nicol\`{o}, J.~Renn, and A.~Steinmann.
\newblock On the massive sine-{G}ordon equation in all regions of collapse.
\newblock {\em Commun. Math. Phys.}, 105:291--326, 1986.
\bibitem{SkTaFa80}
E.K. Sklyanin, L.~A. Takhtadzhyan, and L.~D. Faddeev.
\newblock Quantum inverse problem method {I}.
\newblock {\em Theor. Math. Phys.}, 40:688--706, 1980.
\bibitem{Zam79}
A.~A. Zamolodchikov.
\newblock Factorized {S-Matrices} in two-dimensions as the exact solutions of
certain relativistic quantum field theory models.
\newblock {\em Ann. Phys.}, 120, 1979.
\end{thebibliography}
\end{document}
**