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

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.

Progress on the study of the Ginibre ensembles II: GinOE and GinSE

“…For a fixed Q, the aymptotic expansion of the partition function Z C N (NQ) in (5.5) was discussed in [38, §5.3] in detail. For radially symmetric potentials, the use of (5.52) and (5.13) was made in [40] to show that 1), (5.56) where (5.57)…”