94-343 Hurd T.R.

Charge correlations for the two dimensional Coulomb gas (40K, Latex) 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/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}.

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