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

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 − 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.

Exponential moments for disk counting statistics at the hard edge of random normal matrices

“…√ n in the three other regimes (actually, a similar statement also holds for the higher order cumulants, as can be seen by comparing Corollary 1.5 with Corollary 1.8 and [31,Corollary 1.5]). This indicates that the counting statistics near a hard edge is considerably wilder than near a soft edge, in the bulk or near a semi-hard edge.…”

“…As can be seen from (1.17)-(1.19), the counting statistics in the hard edge regime is drastically different from the counting statistics in the bulk and semi-hard edge regimes (and also very different from the counting statistics in the soft edge regime [28,31]). Indeed, at the hard edge the subleading term is proportional to ln n, while in all other regimes it is proportional to √ n. Furthermore, in the hard edge regime, the leading coefficient C 1 will be shown to depend on the parameters u 1 , .…”

“…In recent years, a lot of works dealing with the counting statistics of two dimensional point processes have appeared [60,25,57,58,43,45,73,28,74,2,31], see also [72] for an earlier work. A common feature of these works is that they all deal exclusively with either "the bulk regime" or with "the soft edge regime".…”

On the characteristic polynomial of the eigenvalue moduli of random normal matrices

“…For a = 0, (1.2) is the moment generating function of the disk counting function E e u π Im ln pn(r) = E e u #{zj :|zj|<r} , (1.5) and in this case w is discontinuous along a circle but has no "circular" root-type singularity. Counting statistics of two-dimensional point processes have attracted a lot of interest in recent years [45,15,43,44,33,31,56,57,17,1,22]. The first two terms in the large n asymptotics of (1.5) were derived in [15,44] 1 .…”

Universality of the number variance in rotational invariant two-dimensional Coulomb gases

“…Indeed, [18] contains precise large-N expansions of all higher cumulants of N a for the complex Mittag-Leffler ensemble. We also refer to [16,20] for a generalisation involving circular-root and merging type singularities.…”