Large gap probabilities of complex and symplectic spherical ensembles with point charges

TL;DR

This paper derives precise asymptotic formulas for the probability of large gaps in eigenvalues of complex and symplectic spherical ensembles, modeled as Coulomb gases with point charges on the Riemann sphere.

Contribution

It provides the second example of exact large gap asymptotics for 2D point processes, explicitly determining the coefficients in the asymptotic expansion.

Findings

Asymptotic probability of no eigenvalues in spherical caps derived

Explicit formulas for coefficients in the asymptotic expansion provided

Extends understanding of large gap probabilities in 2D Coulomb gases

Abstract

We consider $n$ eigenvalues of complex and symplectic induced spherical ensembles, which can be realised as two-dimensional determinantal and Pfaffian Coulomb gases on the Riemann sphere under the insertion of point charges. For both cases, we show that the probability that there are no eigenvalues in a spherical cap around the poles has an asymptotic behaviour as $n\to \infty$ of the form $$ \exp\Big( c_1 n^2 + c_2 n\log n + c_3 n + c_4 \sqrt n + c_5 \log n + c_6 + \mathcal{O}(n^{-\frac1{12}}) \Big) $$ and determine the coefficients explicitly. Our results provide the second example of precise (up to and including the constant term) large gap asymptotic behaviours for two-dimensional point processes, following a recent breakthrough by Charlier.