Universality for random matrices with equi-spaced external source: a case study of a biorthogonal ensemble

TL;DR

This paper establishes the universality of eigenvalue distributions at the edges and in the bulk for a class of random Hermitian matrices with equi-spaced external sources, using novel methods that do not rely on standard formulas.

Contribution

It introduces a new approach to prove universality for biorthogonal ensembles without using Christoffel-Darboux or double-contour formulas.

Findings

Proves edge and bulk universality for the model.

Develops a method applicable to general biorthogonal ensembles.

Shows the potential to handle universality in non-standard determinantal processes.

Abstract

We prove the edge and bulk universality of random Hermitian matrices with equi-spaced external source. One feature of our method is that we use neither a Christoffel-Darboux type formula, nor a double-contour formula, which are standard methods to prove universality results for exactly solvable models. This matrix model is an example of a biorthogonal ensemble, which is a special kind of determinantal point process whose kernel generally does not have a Christoffel-Darboux type formula or double-contour representation. Our methods may showcase how to handle universality problems for biorthogonal ensembles in general.