Dyson gas on a curved contour

Abstract

We introduce and study a model of a logarithmic gas with inverse temperature β on an arbitrary smooth closed contour in the plane. This model generalizes Dyson’s gas (the β-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.