Geometric Expansion of the Log-Partition Function for the Ginibre Gas Obeying Maxwelltire Boltzmann Statistics

S. Poghosyan, H. Zessin

2001, v.7, Issue 4, 581-593


The following problem is discussed for a system of interacting Brownian loops in a bounded domain of $\r^{\nu}$: given the energy ${\cal U}^{\phi}$ of a system, defined in a natural way by means of a stable potential $\phi$ with nice decay properties, the associated log-partition function, $\ln\,Z(\Lambda ,z)$, where $Z(\Lambda ,z)=\int \exp\{-{\cal U}^{\phi}(\mu)\} W_{z\rho _{\Lambda}}(d\mu )$, can be expanded as a function of the geometric characteristics of $\Lambda$, like volume, surface measure etc., if $z>0$ is small enough. ($W_{z\rho%_{\Lambda}}$ denotes the natural reference measure on the configurations of finitely many loops living completely in $\Lambda$ and having ``activity'' $z$.) The constants appearing in this expansion, are uniquely determined by and explicitly represented in terms of $\phi$. The first one can be interpreted as the pressure and the second as the surface tension.

Keywords: partition function,pressure,Ursell function,geometric expansion


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