Exponential moments for disk counting statistics of random normal matrices in the critical regime

TL;DR

This paper derives large $n$ asymptotics for the disk counting statistics in the Mittag-Leffler ensemble at the critical merging regime, revealing central limit theorems and detailed cumulant asymptotics.

Contribution

It provides the first detailed large $n$ asymptotics for the moment generating function of disk counting in the critical regime of the Mittag-Leffler ensemble, including joint cumulants.

Findings

Central limit theorems for disk counting statistics

Asymptotics of joint cumulants and covariances

Large $n$ asymptotics for determinants with merging discontinuities

Abstract

We obtain large $n$ asymptotics for the $m$-point moment generating function of the disk counting statistics of the Mittag-Leffler ensemble. We focus on the critical regime where all disk boundaries are merging at speed $n^{-\smash{\frac{1}{2}}}$, either in the bulk or at the edge. As corollaries, we obtain two central limit theorems and precise large $n$ asymptotics of all joint cumulants (such as the covariance) of the disk counting function. Our results can also be seen as large $n$ asymptotics for $n\times n$ determinants with merging planar discontinuities.