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

We obtain large n asymptotics for the m-point moment generating function of the disk counting statistics of the Mittag-Leffler ensemble, where n is the number of points of the process and m is arbitrary but fixed. We focus on the critical regime where all disk boundaries are merging at speed n − 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 × n determinants with merging planar discontinuities.