The Limiting Characteristic Polynomial of Classical Random Matrix Ensembles

Abstract

We demonstrate the convergence of the characteristic polynomial of several random matrix ensembles to a limiting universal function, at the microscopic scale. The random matrix ensembles we treat are classical compact groups and the Gaussian Unitary Ensemble.In fact, the result is the by-product of a general limit theorem for the convergence of random entire functions whose zeros present a simple regularity property.

Characteristic polynomials of modified permutation matrices at microscopic scale

“…This is exactly the property given in Definition 2.4 in [3] applied to the functions Φ n = ξ n,α for n ≥ 1: they are said to be in probability compact-equicontinuous. Finally, Lemma 2.5 of [3] allows us to conclude that conditionally on y, ξ n,α converges also in law in the topology of uniform convergence on compact sets. In other words, for all bounded functional F from C(C,C) to C, continuous with respect to the topology of the uniform convergence on compact sets,…”

“…With the coupling of virtual isometries introduced by Bourgade, Najnudel and Nikeghbali [2], the authors get an almost sure convergence. Chhaibi, Hovhannisyan, Najnudel, Nikeghbali, and Rodgers [3] extend the study to the special orthogonal group, the symplectic group, and give a related result for the Gaussian Unitary Ensemble.…”

Operator level limit of the circular Jacobi $β$-ensemble

“…ds, and AC([0, 1)) is the set of absolutely continuous real functions on [0, 1). We will use the notations Dir(R, u 0 , u 1 ) or Dir(x + iy, u 0 , u 1 ) for the the operator τ defined via (7) and (8) with boundary conditions u 0 , u 1 on the domain (11). We sometimes replace the R 2 vector by the element in R ∪ {∞} corresponding to the ratio of its two coordinates:…”

“…We review the construction given Section 3 of [35] that shows how a finitely supported probability measure on the unit circle can be represented using a Dirac operator of the form (7). (See also Section 5 of [34].…”

“…[11] provides corrections to these results in the case when β is an even integer or equal to 1. Scaling limits of characteristic polynomials of classical random matrix ensembles were also studied in [8] and [7].…”