 94343 Hurd T.R.
 Charge correlations for the two dimensional Coulomb gas
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Oct 31, 94

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Abstract. This paper is a summary of mathematical results contained in
\cite{Hur94c} concerning integer charge correlations for the Coulomb
gas/sineGordon 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 $\beta<\beta_m=8\pi(11/m)$.
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 KosterlitzThouless 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
$\cO(z^{1/(\beta/4\pi2)})$ the free field power law
$xy^{\beta/2\pi}$ at long distances crosses over to a slower power law
$xy^{4}$. This verifies a conjecture of Fr\"{o}hlich and Spencer
\cite{FrSp80}.
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