Exact solution for mean energy of 2d Dyson gas at beta = 1

TL;DR

This paper derives an exact finite-N solution for the mean energy of a 2D Dyson gas at beta=1, revealing its asymptotic behavior and the presence of half-integer powers in the expansion.

Contribution

It provides the first exact finite-N expression for the mean energy at beta=1 using hypergeometric functions and analyzes its large-N asymptotics.

Findings

Exact finite-N solution in closed form

Asymptotic expansion with half-integer powers

Recursion relation for mean energy

Abstract

Mean Coulomb energy of 2d Dyson gas in quadratic potential is examined from combinatorial viewpoint. For beta = 1, we find a recursive relation on mean energy and obtain its exact (finite N) solution in closed form in terms of the hypergeometric function 3F2. Using this exact solution, we derive the large-N asymptotic expansion of mean energy and show, that this expansion contains half-integer powers of N.