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

Abstract

AbstractWe obtain largenasymptotics for them-point moment generating function of the disk counting statistics of the Mittag–Leffler ensemble, wherenis the number of points of the process andmis arbitrary but fixed. We focus on the critical regime where all disk boundaries are merging at speedn−12, either in the bulk or at the edge. As corollaries, we obtain two central limit theorems and precise largenasymptotics of all joint cumulants (such as the covariance) of the disk counting function. Our results can also be seen as largenasymptotics forn × ndeterminants with merging planar discontinuities.