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

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.