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\begin{titlepage}
\vspace*{3cm}
\begin{center}
{\Large\bf Dyson's Model of Interacting Brownian Motions\bigskip\\
at Arbitrary Coupling Strength}\bigskip\bigskip\bigskip\\
{\large Herbert Spohn}\bigskip\\
Theoretische Physik, Ludwig-Maximilians-Universit\"at\\
Theresienstr. 37, D - 80333 M\"unchen, Germany\\
email: {\tt spohn@stat.physik.uni-muenchen.de}
\end{center}
\vspace{6cm}
\noindent
{\bf Abstract.} For Dyson's model of Brownian motions we prove that
the fluctuations are of order one and, in a scaling limit, are governed
by an infinite dimensional Ornstein-Uhlenbeck process. This extends a
previous result valid only at the free Fermion point $\beta = 2$.
Dyson's model can also be interpreted as a random surface.
Our result implies that the surface statistics is governed by a
massless Gaussian field in the scaling limit.\end{titlepage}
\section{Introduction}
\setcounter{equation}{0}
We consider $N$ ``particles'' with positions
$x_{1}(t),\ldots,x_{N}(t)$ at time $t$ on the circle $[0,\ell]$.
These particles diffuse and are governed by the stochastic
differential equations
\begin{equation}
dx_{j}(t) = a_{j}(x(t))dt + \sqrt{2} db_{j}(t), \quad j=1,\ldots, N,
\end{equation}
with $\{ b_{j}(t), j=1,\ldots,N\}$ independent standard Brownian motions
and with the drift
\begin{equation}
a_{j}(x) = \beta \sum_{i=1, i \neq j}^{N}{\pi \over \ell} \cot
(\pi (x_{j} - x_{i})/\ell), \quad \beta > 0.
\end{equation}
The
diffusion process $x(t) = (x_{1}(t),\ldots,x_{N}(t))$ is assumed to
be stationary. Then $x(t)$ is a reversible process with stationary
measure
\begin{equation}
|\psi(x)|^{2} = Z^{-1} \prod_{i < j=1}^{N}|\sin (\pi (x_{i} -
x_{j})/\ell)|^{\beta}.
\end{equation}
The model (1.1) was introduced by Dyson in 1962 \cite{D} in the
context of random matrices. He considered $N\times N$ real orthogonal
and unitary random matrices. Their eigenvalues $\exp[i 2\pi
x_{j}/\ell],j=1,\ldots,N$, have distribution $|\psi(x)|^{2}$
for $\beta = 1$, resp. for $\beta =2$. The eigenvalues are very
stiffly arranged and have small fluctuations. The distance between
neighboring eigenvalues has a distribution which vanishes at the
origin. In constructing the reversible diffusion process (1.1)
one obtains a dynamical interpretation of this level repulsion.
The case $\beta = 2$ has been studied before \cite{S1,So}. We
generalize here to arbitrary $\beta > 0$, which requires a
novel technique, since the free Fermion method of $\beta = 2$ is no longer
applicable.
The generator for (1.1) reads, with $\partial_{j}= \partial/\partial x_{j}$,
\begin{equation}
L_{N}= \sum_{j=1}^{N}a_{j}(x)\partial_{j} +
\sum_{j=1}^{N}\partial_{j}^{2}.
\end{equation}
Since $L_{N}$ is reversible, we can associate to it the self-adjoint
hamiltonian $H_{N}$ by
\begin{equation}
- \psi L_{N} \psi^{-1} = H_{N}- E_{N},
\end{equation}
which turns out
to be the famous Calogero-Sutherland hamiltonian \cite{Su1,Su2}
\begin{equation}
H_{N}= - \sum_{j=1}^{N}\partial_{j}^{2} + {\pi^{2}\beta(\beta -2)
\over 2 \ell^{2}}\sum_{i -1$. For $N > 2$ to have a square integrable
ground state requires $\beta > - 2/N$. In the interval $ 1 <
\beta < 2 + (2/N)$ $H_{N}$ has two self-adjoint extensions with a
positive transition kernel. These are labeled by $\beta$ and $2 - \beta$ with
corresponding ground state as defined in (1.3).
For our enterprise the stochastic differential equations (1.1) will
be more natural. We require $\beta > 0$. Then $|\psi(x)|^{2}$ is
normalizable for all $N$. Let $\Omega = \{x, x_{i} = x_{j}, i \neq
j, i,j = 1,\ldots,N\}$. We start the process at $x \in
[0,\ell]^{N} \setminus \Omega$, since $\Omega$ has $|\psi|^{2}$-
measure zero. To reach a point of $\Omega$ where three or more
particles coincide has probability zero.
The probability for two particles to coincide depends on $\beta$.
Close to $\Omega$ the drift
$a_{j}$ diverges as $\beta (x_{j} - x_{i})^{-1}$. Thus the relative
distance between two particles behaves as the Bessel process of order
$(\beta -1)/2$ for small separation. We conclude that if $\beta \geq
1$, then with probability one $\Omega$ is never reached. Thus for
all $t$ we have $x_{i}(t) \neq x_{j}(t)$, $ i \neq j$. There is no
boundary condition at $\Omega$. If $0 < \beta < 1$, then $x(t)$ will
hit $\Omega$ with a non-zero probability and
$x_{i}(t) = x_{j}(t)$ at some time $t$ for some pair $ (i ,j)$.
Since the solution to
(1.1) is well-defined even for $x(0) \in \Omega$, we can continue
either keeping the order (reflecting boundary conditions) or
interchanging the order (transmitting boundary conditions) of the
pair of particles. Later on we will consider only functions symmetric
in the particle label. Thus either convention is fine. However,
as to be explained in the following paragraph, for the
surface picture reflecting boundary conditions are
more natural. Thus at time $t = 0$ we label the particles such
that $0 \leq x_{1}(0) < \ldots < x_{N}(0) < \ell$. Then almost surely
(1.1) has a
unique solution with continuous sample paths and at any time $t$ we
have
\begin{equation}
x_{1}(t) \leq x_{2}(t) \leq \ldots \leq x_{N}(t) \quad {\rm mod} \; \ell.
\end{equation}
We refer to \cite{RW} for a more detailed discussion.
The hamiltonian (1.6) suggests a physical application
of interest \cite{L,S3}. We regard
$t \mapsto x_{j}(t)$ as a step of unit height in a vicinal
surface. The height function, $h$, of the surface takes then
integer values. We define $h$ first for $t=0$ by
\begin{equation}
h(x,0) = \left\{ \begin{array}{ll}
0 & {\rm for}\;\; 0 \leq x < x_{1}(0),\\
j & {\rm for}\;\; x_{j}(0) \leq x < x_{j+1}(0) \; , j = 1,\ldots,N-1,\\
N & {\rm for}\;\; x_{N}(0) \leq x < \ell
\end{array}
\right.
\end{equation}
and extend $h$ to all $x \in \Bbb{R}$ by
$h(x + n \ell,0) = h(x,0) + n N$ with integer $n$. Let
$J([t_{1},t_{2}])$ be the current through the origin integrated over the
time span $[t_{1},t_{2}]$, i.e. $J([t_{1},t_{2}])$ is the number of
particles crossing forwards minus those crossing backwards the origin
during $[t_{1},t_{2}]$. Then for all $x,t \in {\Bbb{R}}^{2}$
\begin{equation}
h(x,t) = \left\{ \begin{array}{ll}
h(x,0) - J([0,t]) &{\rm for}\;\; t > 0, \\
h(x,0) + J([t,0]) &{\rm for}\;\; t < 0.
\end{array}
\right.
\end{equation}
In the
transverse ($x$) direction the surface has period $\ell$ and a miscut
angle determined through the step density $\rho = N/\ell$.
For the physical interpretation we associate to $H_{N}$
the formal Boltzmann weight
\begin{eqnarray}
\exp &\left[ - \sum_{j=1}^{N}\int \dot{x}_{j}(t)^{2}dt
- {\pi^{2}\beta(\beta -2)
\over 2 \ell^{2}}\sum_{i 0$ we define the fluctuation field
\begin{equation}
\xi^{N}(f,t) = \sum_{j=1}^{N}f(x_{j}(t)).
\end{equation}
By definition $\Bbb{E}$$_{N}(\xi^{N}(f,t)) = N \hat{f}_{0} = 0$. Note
that the prefactor in (2.5) is one. This reflects that
fluctuations are severely suppressed because of the strong repulsive
forces. By linear duality we regard $t \mapsto \xi^{N}(\cdot,t)$ as
Hilbert space valued stochastic process. As to be shown,
$\xi^{N}(\cdot,t)$
almost surely takes values in ${\cal
H}_{-3 - \kappa}$ and is weakly continuous in
$t$. Let $Q^{N}$ be the path measure on
$C(\Bbb{R}$$,{\cal H}_{-3 - \kappa})$ induced by $P^{N}$.
In the limit $N \rightarrow \infty$, $Q^{N}$ is well approximated
by an infinite-dimensional Ornstein-Uhlenbeck process $\xi(f,t)$,
which is governed by the stochastic partial differential equation
\begin{equation}
d\xi(f,t) = \xi(- (\beta/2) \sqrt{ - (\partial /\partial x)^{2}}f,t)dt
+ dW(f',t)
\end{equation}
with $f \in {\cal H}_{3 + \kappa}$ and $dW(f,t)$ white noise
normalized as $\Bbb{E}$$(dW(f,t)dW(g,s)) = 2 \delta (t-s) dt ds(1/2\pi)
\int_{0}^{2\pi}f(x)g(x)dx$. The stationary measure for $\xi(f,t)$
is Gaussian with covariance
\begin{equation}
{\Bbb{E}}(\xi(f,t)\xi(g,t)) = {2 \over \beta}
\sum_{k} |k|\hat{f}_{k}^{\ast}\hat{g}_{k}.
\end{equation}
We denote by $Q$ the path measure on $C(\Bbb{R}$$,{\cal H}_{-3 - \kappa})$
of the stationary Ornstein-Uhlenbeck process (2.6).
As our main result we state\bigskip\\
{\bf Theorem 1}. {\it Let $Q^{N}$ be the induced path measure for
(2.5) and
let $Q$ be the path measure
of the stationary Ornstein-Uhlenbeck process (2.6)
on $C(\Bbb{R}$$,{\cal H}_{-3 - \kappa})$. Then in the sense of
weak convergence of measures}
\begin{equation}
\lim_{N \rightarrow \infty} Q^{N} \; = \; Q .
\end{equation}
\bigskip\\
{\bf Remarks}. (i) In the surface picture the steps are the $x$-derivative of the
surface height $h(x,t)$, cf. Eq. (1.8). If we scale the surface
according to (2.1), then the step height is $1/N$ and the average
surface $ = x/ 2\pi$. Theorem 1 (together with a bound on the large
values of $\xi^{N}$) implies that
\begin{equation}
\lim_{N \rightarrow \infty}{\Bbb{E}}_{N}(\xi^{N}(f,t)\xi^{N}(g,0))
= {2 \over \beta}
\sum_{k}\hat{f}_{k}^{\ast}%\hat{g}_{k}|k|e^{-(\beta/2) |k||t|}.
\end{equation}
Therefore the surface
fluctuations, $h^{N}(x,t) - x/ 2\pi$, are of order $1/N$ and
in the limit $N
\rightarrow \infty$ are Gaussian with covariance
$8/(\beta^{2} k^{2} + 4 \omega^{2})$ in Fourier space representation with respect
to $x,t$. This means that the surface has a large scale statistics
governed by a massless free field,
which is the generally expected behavior if the surface tension depends
smoothly on the slope. In our case the surface tension is the
ground state energy $e(\rho)$ per unit volume of the
Calogero-Sutherland model. By (1.7) $e(\rho) = \beta^{2}\pi^{2}\rho^{3}/12$
which satisfies the hypothesis on smooth dependence on the slope.
The massles Gaussian limit has been
established only for a few surface models, amongst them
the Ginzburg-Landau $\nabla \phi$
surface model \cite{NS,GOS}. \medskip\\
(ii) One would like to take first the limit $\ell,N \rightarrow \infty$,
$ \rho = N/\ell$ fixed, in (1.1). The force between the Brownian
particles is then repulsive and given by $\beta/(x_{j} - x_{i})$.
Except for $\beta = 2$, one does not even know the existence of the
infinite volume stationary measure, which is a prerequisite for
constructing the process. Also the Johansson
result used below is not available in this setting.\medskip\\
(iii) Over the years there has been a considerable effort to compute
ground state correlations in the Calogero-Sutherland model. In our
context the results on the density-density correlations are of
interest. On the microscopic scale (1.1) let
\begin{equation}
\rho(x,t) = \sum_{i=1}^{N}\delta(x - x_{j}(t))
\end{equation}
and let
\begin{equation}
<\rho (x,t)\rho (0,0)>_{ N,\ell} -
<\rho (0,0)>_{ N,\ell}^{2}= S_{N,\ell}(x,t),
\end{equation}
average in the stationary process. For rational $\beta = p/q$ the
infinite volume limit
\begin{equation}
S_{\rho}(x,t) = \lim_{N,\ell \rightarrow \infty, N/\ell = \rho}
S_{N,\ell}(x,t)
\end{equation}
is computed explicitely in \cite{Ha} and expressed as a $p + q$-fold
Dotsenko-Fateev integral. The asymptotics of these integrals are
studied in \cite{Fo} with the result
\begin{equation}
S_{\rho}(x,t) = {1 \over 2 \beta \pi^{2}}((x - i\pi\rho\beta
|t|)^{-1} +
(x + i\pi\rho\beta |t|)^{-1})
\end{equation}
for large $x,t$. Defining the structure function
\begin{equation}
\hat{S}_{\rho}(k,t) = \int e^{ikx} S_{\rho}(x,t) dx,
\end{equation}
we obtain
\begin{equation}
\hat{S}_{\rho}(k,t) = {1 \over \pi\beta} |k| e ^{-\pi \rho \beta
|k||t|}.
\end{equation}
As expected this result agrees with (2.8), except for the prefactor
$1/2 \pi$ which results from the conventions in defining Fourier
transforms.
To us it seems prohibitively complicated to establish a Gaussian limit
with similar kind of techniques.\medskip\\
(vi) Physically it would be of great interest to prove Theorem 1 for
a diffusion process where in the Hamiltonian $H_{N}$ the long range
potential $1/x^{2}$ is replaced by a short range potential
and where Dirichlet boundary conditions are imposed on the crossing set
$\Omega$.\medskip\\
(v) If the drift is given by
\begin{equation}
a_{j}(x) = - \sum_{i=1, i \neq j}^{N} \partial_{j}U(x_{j} - x_{i})
\end{equation}
with a sufficiently weak finite range potential $U$, then a result
analogous to Theorem 1 has been proved in \cite{S4}. If we reintroduce
the density in (2.6), then in both cases the noise strength is
$\sqrt{ \rho}$. However the drift term is now local and given by $D
\partial^{2}/\partial x^{2}$ with $ D = \rho/\chi$ and $\chi$ the
static compressibility.
\section{Convergence to an Ornstein-Uhlenbeck Process}
\setcounter{equation}{0}
We follow the standard route of hydrodynamic fluctuation theory
\cite{S2}. We have
\begin{equation}
\xi^{N}(f,t) = \xi^{N}(f,0) + \int_{0}^{t} \gamma^{N}(f)(x(s)) ds
+ M^{N}(f,t).
\end{equation}
$M^{N}(f,t)$ is a martingale and
\begin{equation}
M^{N}(f,t)^{2} - \int_{0}^{t} \gamma_{1}^{N}(f)(x(s)) ds
= M_{1}^{N}(f,t),
\end{equation}
where $M_{1}^{N}(f,t)$ is again a martingale. Here $\gamma^{N}(f)$ and
$\gamma_{1}^{N}(f)$ are given by
\begin{eqnarray}
\gamma^{N}(f)(x) & = & L_{N}\xi^{N}(f)(x) =
\beta {1 \over 2N}\sum_{i = 1,i \neq j}^{N} \cot
((x_{j} - x_{i})/2) f'(x_{j}) + {1 \over N}\sum_{j=1}^{N} f''(x_{j}),
\nonumber \\
\gamma_{1}^{N}(f)(x) & = & (L_{N}\xi^{N}(f)^{2} -
2\xi^{N}(f) L_{N}\xi^{N}(f))(x) =
{2 \over N}\sum_{j=1}^{N} f'(x_{j})^{2}.
\end{eqnarray}
According to Theorem 1 we have to establish that in the limit $N
\rightarrow \infty$ the fluctuation field satisfies
\begin{equation}
\xi(f,t) = \xi(f,0) + \int_{0}^{t} \xi(- (\beta/2)
\sqrt{-(\partial^{2}/\partial x^{2})}f,s) ds
+ W(f',t).
\end{equation}
To our own surprise, the proof requires only static estimates which
we state first. Let $\{x_{j},j=1,\ldots,N\}$ be distributed according to
(2.3). We have\medskip\\
{\bf Lemma 1}. {\it Let $f: [0,2\pi] \rightarrow \Bbb{R}$ be
continuous. Then in distribution}
\begin{equation}
\lim_{N\rightarrow \infty} {1 \over N} \sum_{j=1}^{N}f(x_{j})
= \hat{f}_{0}.
\end{equation}
{\bf Proof}: \cite{BP,KS}. $\Box$ \medskip\\
{\bf Lemma 2}. {\it Let $f \in {\cal H}_{1}$. Then}
\begin{equation}
\lim_{N \rightarrow \infty} {\Bbb{E}}_{N}(\xi^{N}(f)^{2}) =
{2 \over \beta} \sum_{k}|k||\hat{f}_{k}|^{2}.
\end{equation}
{\bf Proof}: In \cite{Jo1,Jo2} Johansson proves
\begin{equation}
\lim_{N \rightarrow \infty} {\Bbb{E}}_{N}(\exp [\xi^{N}(f)]) =
\exp[ {1 \over \beta} \sum_{k}|k||\hat{f}_{k}|^{2}],
\end{equation}
which means that the static fluctuations of $\xi^{N}(f)$ are Gaussian
in the limit.
In \cite{Jo1} only the case $\beta = 2$ is stated. The proof extends to
all $\beta$ as explained in \cite{Jo2}, where the more difficult case
of a non-uniform background \medskip density is handled. $\Box$
With such an input we have \medskip\\
{\bf Lemma 3}. {\it For $f \in {\cal H}_{2}$}
\begin{equation}
\lim_{N \rightarrow \infty} \gamma_{1}^{N}(f) =
2 \sum_{k}k^{2}|\hat{f}_{k}|^{2}
\end{equation}
{\it in distribution}. \medskip\\
{\bf Lemma 4}. {\it Let $f\in {\cal H}_{4}$ and let $A$ be the
integral operator defined by the Hilbert transform}
\begin{equation}
Af(x) = {\beta \over 4 \pi} \int_{0}^{2\pi} \cot ((y-x)/2)f'(y)dy
\end{equation}
{\it in the sense of the Cauchy principal part. Then}
\begin{equation}
\lim_{N \rightarrow \infty} {\Bbb{E}}_{N}((\gamma^{N}(f) - \xi^{N}(Af))^{2}) =
0.
\end{equation} \medskip
{\bf Proof}: We work out the square and
use that
\begin{eqnarray}
\widehat{Af}_{k} & = &{\beta \over 2} (2\pi)^{-2} \int_{0}^{2\pi}\int_{0}^{2\pi}
e^{-ikx} \cot ((y-x)/2)f'(y)dxdy
\nonumber\\
& = & - (\beta/2) |k| \hat{f}_{k}.
\end{eqnarray}
(i) First term. By Lemma 2 and (3.11)
\begin{equation}
\lim_{N \rightarrow \infty} {\Bbb{E}}_{N}(\xi^{N}(Af)^{2}) =
{2 \over \beta} \sum_{k}|k||\widehat{Af}_{k}|^{2}
= { \beta \over 2}\sum_{k}|k|^{3}|\hat{f}_{k}|^{2}.
\end{equation}
(ii) Second term. We use that $\partial_{j}|\psi_{N}(x)|^{2}
= N a^{N}_{j}(x)|\psi_{N}(x)|^{2}$ and partially integrate. Then
\begin{eqnarray}
& - & 2 {\Bbb{E}}_{N}(\gamma^{N}(f)\xi^{N}(Af)) \nonumber \\
& = & - 2 \sum_{i,j=1}^{N} \left\{ {\Bbb{E}}_{N}[a_{j}^{N}(x)f'(x_{j})Af(x_{i})]
+ {1 \over N}{\Bbb{E}}_{N}
[f''(x_{j})Af(x_{i})] \right\} \nonumber \\
& = & {2 \over N} \sum_{i,j=1}^{N} \left\{
{\Bbb{E}}_{N}[\partial_{j}(f'(x_{j})Af(x_{i})]
- {\Bbb{E}}_{N}
[f''(x_{j})Af(x_{i})] \right\} \nonumber \\
& = &
{2 \over N}{\Bbb{E}}_{N}[\sum_{j=1}^{N}f'(x_{j})Af'(x_{j})],
\end{eqnarray}
which by Lemma 1 converges to
\begin{equation}
- \beta \sum_{k}|k|^{3}|\hat{f}_{k}|^{2}.
\end{equation}
(iii) Third term. As before we use partial integration,
\begin{eqnarray}
& & {\Bbb{E}}_{N}(\gamma^{N}(f)^{2}) \nonumber \\
& = & \sum_{i,j=1}^{N} \left\{
{\Bbb{E}}_{N}[a_{i}^{N}(x)f'(x_{i})a_{j}^{N}(x)f'(x_{j})]
+ {2 \over N}
{\Bbb{E}}_{N}[a_{i}^{N}(x)f'(x_{i})f''(x_{j})] \right. \nonumber \\
& & \left. + {1 \over N^{2}}
{\Bbb{E}}_{N}[f''(x_{i})f''(x_{j})] \right\} \nonumber \\
& = & {1 \over N^{2}}
\sum_{i,j=1}^{N} \left\{
{\Bbb{E}}_{N}[\partial_{i}\partial_{j}(f'(x_{i})f'(x_{j}))
- N(\partial_{j}a_{i}^{N})(x)f'(x_{i})f'(x_{j})] \right. \nonumber \\
& & \left. -2 {\Bbb{E}}_{N}[\partial_{i}(f'(x_{i})f''(x_{j}))]
+ {\Bbb{E}}_{N}[f''(x_{i})f''(x_{j})] \right\} \\
& = &
{1 \over N^{2}}{\Bbb{E}}_{N}[\sum_{j=1}^{N}f''(x_{j})^{2}] +
{\beta \over 8 N^{2}} {\Bbb{E}}_{N}[\sum_{i,j=1}^{N}
(\sin((x_{i} - x_{j})/2))^{-2}(f'(x_{i}) - f'(x_{j}) )^{2}]. \nonumber
\end{eqnarray}
The first summand vanishes as $N \rightarrow \infty$ by the law of
large numbers (2.4) and since $f \in {\cal H}_{4}$. Again by the law of
large numbers the second
summand converges to
\begin{eqnarray}
& &{\beta \over 8} (2\pi)^{-2}
\int_{0}^{2\pi}\int_{0}^{2\pi}
(\sin ((x-y)/2))^{-2}(f'(x) - f'(y))^{2}dxdy
\nonumber \\
& = &{\beta \over 2} \sum_{k}|k|^{3}|\hat{f}_{k}|^{2}.
\end{eqnarray}
We conclude that the limit in (3.10) vanishes. $\Box$ \medskip
Next we have to establish support and tightness of $Q^{N}$. We
denote by $||f||_{\alpha}$ the norm in ${\cal H_{\alpha}}$, cf.
definition above (1.12). \medskip\\
{\bf Lemma 5}. {\it Let $f \in {\cal H}_{2}.$ Then}
\begin{equation}
Q^{N}(\sup_{0 \leq t \leq T}|\xi^{N}(f,t)|^{2}) \leq
c_{1}T \sum_{k}|k|^{2}|\hat{f}_{k}|^{2}.
\end{equation}
{\bf Proof}: For any stationary Markov process, $X_{t}$, with generator $L$ and
invariant measure $\mu$ we have
\begin{equation}
{\Bbb{E}}(\sup_{0 \leq t \leq T}|G(X_{t})|^{2}) \leq 3
{\Bbb{E}}(|G|^{2}) -56 T{\Bbb{E}}(GLG)
\end{equation}
for $G \in L^{2}(\mu) \cap H_{1}(\mu)$ \cite{OY}.
The assertion follows from Lemma 1 and
from $||f||_{1} \leq ||f||_{2}$ . $\Box$ \medskip\\
{\bf Lemma 6}. {\it $Q^{N}$ and $Q$ are supported by
$C({\Bbb{R}},{\cal H}_{-3 - \kappa})$ for some $\kappa > 0$.} \medskip\\
{\bf Proof}: We have as distribution
\begin{equation}
\xi^{N}(x,t) = \sum_{j=1}^{N}\delta (x_{j}(t) -x) - {N \over 2\pi}.
\end{equation}
Therefore
\begin{equation}
\hat{\xi}_{k}^{N}(t) = {1 \over 2\pi}\sum_{j=1}^{N}\exp[-ikx_{j}(t)], \;
k\neq 0, \; \hat{\xi}_{0}^{N}(t) = 0,
\end{equation}
and
\begin{eqnarray}
& &{\Bbb{E}}_{N}(\sup_{0 \leq t \leq T}||\xi(\cdot,t) ||_{-3 - \kappa}^{2})
\leq \sum_{k \neq 0}|k|^{-3 - \kappa}
{\Bbb{E}}_{N}(\sup_{0 \leq t \leq T}|\hat{\xi}_{k}(t)|^{2})
\nonumber \\
& \leq & c' \sum_{k \neq 0}|k|^{-3 - \kappa}|k|^{2} < \infty
\end{eqnarray}
by Lemma 5. The continuity of $t \mapsto \xi^{N}(f,t)$
and $t \mapsto \xi(f,t)$ can be checked directly, since $f \in {\cal
H}_{3+\kappa}$. $\Box$ \medskip\\
{\bf Lemma 7}. {\it The family of measures $\{Q^{N}, N = 1,2,\ldots\}$ is
tight.}\medskip\\
{\bf Proof}: The proof is identical to \cite{S5}, page 207. $\Box$\medskip\\
{\bf Proof of Theorem 1}: Since the family $\{Q^{N}, N = 1,2,\ldots\}$
is tight, we can choose a convergent subsequence with limit point $Q$.
By Lemma 4 we conclude that for $f \in {\cal H}_{4}$
\begin{equation}
M(f,t) = \xi(f,t) - \xi(f,0) - \int_{0}^{t} \xi(- (\beta /2)
\sqrt{-(\partial^{2}/\partial x^{2})}f,s) ds
\end{equation}
is a martingale under $Q$. For $f \in {\cal H}_{4} \subset
{\cal H}_{3}$ by Lemma 3
\begin{equation}
M_{1}(f,t) = M(f,t)^{2} - 2t {1 \over 2\pi}\int_{0}^{2\pi}f'(x)^{2}dx
\end{equation}
is also a martingale under $Q$. The martingale problem (3.22), (3.23)
with $f \in {\cal H}_{4}$ has as
unique solution the infinite dimensional Ornstein-Uhlenbeck process
(2.6) \cite{HS}. Its inital measure is fixed by (3.7). Alternatively
we can use that the Ornstein-Uhlenbeck process (2.6) must be
stationary as a limit of stationary processes. This determines (3.7)
uniquely as its single time distribution. Since the limit measure
is unique, any subsequence has the same limit. $\Box$ \bigskip\\
{\large \textbf{Acknowledgements.}} I thank M. Pr\"{a}hofer, Ya. G.
Sinai, and A. Soshnikov for instructive discussions and M. Kiessling
for a carefull reading of the manuscript.
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\end{document}