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

Abstract

We study the characteristic polynomial pn(x) = n j=1 (|zj| − x) where the zj are drawn from the Mittag-Leffler ensemble, i.e. a two-dimensional determinantal point process which generalizes the Ginibre point process. We obtain precise large n asymptotics for the moment generating function E[e u π Im ln pn(r) e a Re ln pn (r) ], in the case where r is in the bulk, u ∈ R and a ∈ N. This expectation involves an n × n determinant whose weight is supported on the whole complex plane, is rotation-invariant, and has both jump-and root-type singularities along the circle centered at 0 of radius r. This "circular" root-type singularity differs from earlier works on Fisher-Hartwig singularities, and surprisingly yields a new kind of ingredient in the asymptotics, the so-called associated Hermite polynomials.

Partition functions of determinantal and Pfaffian Coulomb gases with radially symmetric potentials

“…We mention that such error estimates also naturally appeared in similar computations, see e.g. [13,19,20]. Nevertheless, we expect that the estimates can be improved with more effort.…”

“…The weight function with a jump-type singularity gives rise to the moment generating function of the disc counting function. It has been extensively studied in recent years [4,8,13,17,20,22]. (We also refer to [26,40,41,51] for physical motivations of these problems from the counting statistics of rotating free fermions.)…”

“…(We also refer to [26,40,41,51] for physical motivations of these problems from the counting statistics of rotating free fermions.) Finally, a root-type singularity arises in the study of the log-characteristic polynomials [13,24,53].…”

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

“…The works [25,58,45,28,31] were already mentioned earlier in the introduction and deal with determinants with discontinuities in dimension two. Determinants corresponding to the logarithmic test-function l z , for some special ensembles, have attracted considerable attention in recent years [77,38,21,20], see also e.g. [13,15,16,18,61].…”

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.…”