Theory of random matrices with strong level confinement: orthogonal polynomial approach

TL;DR

This paper develops an orthogonal polynomial approach to analyze large random matrices with strong level confinement, revealing universal local eigenvalue correlations and explicit spectral density formulas.

Contribution

It introduces a novel polynomial-based method to study non-Gaussian ensembles, deriving asymptotic correlations and universal behaviors independent of confinement details.

Findings

Universal local eigenvalue correlations are established.

Explicit formulas for spectral density and Green's functions are derived.

Global correlations depend only on spectrum endpoints.

Abstract

Strongly non-Gaussian ensembles of large random matrices possessing unitary symmetry and logarithmic level repulsion are studied both in presence and absence of hard edge in their energy spectra. Employing a theory of polynomials orthogonal with respect to exponential weights we calculate with asymptotic accuracy the two-point kernel over all distance scale, and show that in the limit of large dimensions of random matrices the properly rescaled local eigenvalue correlations are independent of level confinement while global smoothed connected correlations depend on confinement potential only through the endpoints of spectrum. We also obtain exact expressions for density of levels, one- and two-point Green's functions, and prove that new universal local relationship exists for suitably normalized and rescaled connected two-point Green's function. Connection between structure of Szeg\"o function entering strong polynomial asymptotics and mean-field equation is traced.