Dyson gas on a curved contour

TL;DR

This paper generalizes Dyson's gas to arbitrary smooth contours, deriving the large N free energy expansion and linking it to spectral determinants of boundary operators, thus connecting statistical mechanics with spectral geometry.

Contribution

It introduces a generalized Dyson gas model on arbitrary contours and computes the large N free energy expansion using loop equations, relating it to spectral determinants of boundary operators.

Findings

Main free energy contribution expressed via spectral determinants.

Connection established between Dyson gas statistics and spectral geometry.

Explicit computation of large N expansion terms.

Abstract

We introduce and study a model of a logarithmic gas with inverse temperature $\beta$ on an arbitrary smooth closed contour in the plane. This model generalizes Dyson's gas (the $\beta$-ensemble) on the unit circle. We compute the non-vanishing terms of the large $N$ expansion of the free energy ($N$ is the number of particles) by iterating the "loop equation" that is the Ward identity with respect to reparametrizations and dilatation of the contour. We show that the main contribution to the free energy is expressed through the spectral determinant of the Neumann jump operator associated with the contour, or equivalently through the Fredholm determinant of the Neumann-Poincare (double layer) operator. This result connects the statistical mechanics of the Dyson gas to the spectral geometry of the interior and exterior domains of the supporting contour.