Hurd T.R.
Charge correlations for the two dimensional Coulomb gas
(40K, Latex)

ABSTRACT.  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}.