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, 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.

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

“…As we show below, the above asymptotic expansions have several interesting consequences; for example, VarOEN.r j / n in the hard edge regime, while VarOEN.r j / p 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 are considerably wilder than near a soft edge, in the bulk or near a semi-hard edge.…”

“…As can be seen from (1.14)-(1.16), the counting statistics in the hard edge regime are 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 p n. Furthermore, in the hard edge regime, the leading coefficient C 1 will be shown to depend on the parameters u 1 ; : : : ; u m in a highly non-trivial non-linear way.…”

“…In recent years, a lot of works dealing with the counting statistics of two-dimensional point processes have appeared [2,25,28,31,43,45,57,58,60,72,73], see also [71] 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.…”

Universality in the number variance and counting statistics of the real and symplectic Ginibre ensemble

“…This physical situation is akin to the well known lowest Landau level problem of electrons in a plane and in the presence of a perpendicular magnetic field [32,45,48]. This connection with the physics of the lowest Landau levels has naturally motivated the study of the FCS in the complex Ginibre ensemble, denoted here as GinUE [1,6,7,14,25,27,28,39,73,90] (for a recent review see [16]), as well as some natural extensions of it, including the higher Landau levels [68,69,91,94], related to the socalled poly-analytic Ginibre ensemble [53]. We also refer to [78,79] and references therein for earlier work on the counting statistics of Hermitian random matrix ensembles and its applications to one-dimensional systems of trapped fermions.…”

Universality of the Number Variance in Rotational Invariant Two-Dimensional Coulomb Gases

“…This physical situation is akin to the well known lowest Landau level problem of electrons in a plane and in the presence of a perpendicular magnetic field [32,45,48]. This connection with the physics of the lowest Landau levels has naturally motivated the study of the FCS in the complex Ginibre ensemble, denoted here as GinUE [1,6,7,14,25,27,28,39,73,88] (for a recent review see [16]), as well as some natural extensions of it, including the higher Landau levels [68,69,89,92], related to the so-called poly-analytic Ginibre ensemble [53]. For such models, analytical progress is possible thanks to the fact that the underlying point processes are DPPs, for which very powerful analytical tools are available, already for a finite number of points N , see [56,59].…”