Almost-Hermitian random matrices and bandlimited point processes

Abstract

We study the distribution of eigenvalues of almost-Hermitian random matrices associated with the classical Gaussian and Laguerre unitary ensembles. In the almost-Hermitian setting, which was pioneered by Fyodorov, Khoruzhenko and Sommers in the case of GUE, the eigenvalues are not confined to the real axis, but instead have imaginary parts which vary within a narrow "band" about the real line, of height proportional to 1 N , where N denotes the size of the matrices. We study vertical cross-sections of the 1-point density as well as microscopic scaling limits, and we compare with other results which have appeared in the literature in recent years. Our approach uses Ward's equation and a property which we call "crosssection convergence", which relates the large-N limit of the cross-sections of the density of eigenvalues with the equilibrium density for the corresponding Hermitian ensemble: the semi-circle law for GUE and the Marchenko-Pastur law for LUE.

Random normal matrices in the almost-circular regime

“…We emphasise that the scaling limit (1.15) agrees with the one appearing in the context of almost-Hermitan random matrices [24][25][26]. We also refer to [3,4] and references therein for some known universality results in this class. (Cf.…”

“…Furthermore, contrary to almost-Hermitian random matrices, where only a few specific models were analysed (see e.g. [1,3,4,12,36]), almostcircular ensembles provide concrete models allowing explicit asymptotic analysis possible.…”

“…This behaviour (1.9) makes the droplet (1.6) close to the unit circle. Furthermore, by (1.9), one can notice that the parameter ρ defined by (1.8) plays the role in describing the width of the droplet, see [4] for more about the geometric meaning of such a parameter. Now we introduce scaling limits of almost-circular ensembles.…”

“…The following theorem is an immediate consequence of [4,Thoerem 3.8]. This approach is based on Ward's equation, see Subsection 2.4 for more about this method.…”

“…The limiting 1-point function determines a unique determinantal point field, see [9, Lemma 1]. We also refer to [8,Theorem 1.1] and [4,Lemma 3.5] for the structure of a limiting correlation kernel of the rescaled system. To be more precise, for a class of potentials which are strictly subharmonic near the droplet, a limiting correlation kernel K of a properly rescaled system is of the form…”