Level Crossing in Random Matrices. III. Analogs of Girko's circular and Wigner's semicircle laws

TL;DR

This paper investigates the asymptotic behavior of level crossings in random matrix pencils across various ensembles, establishing conditions for convergence to deterministic limits and exploring universality and spectral degeneracies.

Contribution

It provides new conditions under which the empirical measure of level crossings converges to a deterministic limit, extending Girko's and Wigner's laws to matrix pencils.

Findings

Convergence of level crossing measures to explicit limits in complex Gaussian ensembles.

Conditional results for i.i.d. ensembles based on universality assumptions.

Real case analysis showing measures do not concentrate on the real line under certain conditions.

Abstract

We study the asymptotic distribution of level crossings for random matrix pencils A_n+\lambda B_n in several ensembles, including complex and real i.i.d. matrices and Gaussian/Hermitian settings. We derive a representation of the normalized log-discriminant in terms of pairwise eigenvalue interactions and formulate conditions under which its limit is governed by a deterministic potential. Under assumptions combining a uniform circular law, logarithmic tail control, and small-spacing (repulsion) estimates, we prove convergence of the empirical measure of level crossings to an explicit deterministic limit. In the complex Gaussian case these assumptions are verified (modulo a uniformity step), while in the general i.i.d. setting the results are conditional and motivated by universality theory. We further analyze the real case, showing that any limiting measure does not concentrate on the real projective line under suitable hypotheses, and discuss analogous phenomena for elliptic/Hermitian ensembles. Our results highlight the role of logarithmic energy and universality in governing spectral degeneracies of random matrix pencils.