Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

TL;DR

This paper derives detailed asymptotic expansions of partition functions for two-dimensional Coulomb gases with radially symmetric potentials, revealing how droplet connectivity influences these formulas and confirming some predictions while introducing new terms.

Contribution

It provides the first detailed asymptotic expansions of the log-partition functions for determinantal and Pfaffian Coulomb gases with radially symmetric potentials, including new $O(N)$-terms for symplectic ensembles.

Findings

Formulas depend on droplet connectivity.

Results agree with Zabrodin and Wiegmann predictions up to a constant.

New logarithmic potential term appears in symplectic ensembles.

Abstract

We consider random normal matrix and planar symplectic ensembles, which can be interpreted as two-dimensional Coulomb gases having determinantal and Pfaffian structures, respectively. For general radially symmetric potentials, we derive the asymptotic expansions of the log-partition functions up to and including the $O(1)$-terms as the number $N$ of particles increases. Notably, our findings stress that the formulas of the $O(\log N)$- and $O(1)$-terms in these expansions depend on the connectivity of the droplet. For random normal matrix ensembles, our formulas agree with the predictions proposed by Zabrodin and Wiegmann up to a universal additive constant. For planar symplectic ensembles, the expansions contain a new kind of ingredient in the $O(N)$-terms, the logarithmic potential evaluated at the origin in addition to the entropy of the ensembles.