Large N expansion for the 2D Dyson gas

TL;DR

This paper develops a 1/N expansion for the free energy of a 2D Dyson gas, connecting random matrix theory, complex analysis, and spectral geometry, and explores implications for field theories and boundary value problems.

Contribution

It introduces a novel 1/N expansion method for the Dyson gas's free energy and links it to complex analysis, spectral geometry, and field theories.

Findings

Derived large N free energy expansion for the Dyson gas.

Connected Dyson gas properties to Dirichlet boundary value problems.

Identified links with bosonic field theory and topological field theories.

Abstract

We discuss the 1/N expansion of the free energy of N logarithmically interacting charges in the plane in an external field. For some particular values of the inverse temperature beta this system is equivalent to the eigenvalue version of certain random matrix models, where it is refered to as the "Dyson gas" of eigenvalues. To find the free energy at large N and the structure of 1/N-corrections, we first use the effective action approach and then confirm the results by solving the loop equation. The results obtained give some new representations of the mathematical objects related to the Dirichlet boundary value problem, complex analysis and spectral geometry of exterior domains. They also suggest interesting links with bosonic field theory on Riemann surfaces, gravitational anomalies and topological field theories.