Proof of universality of the Bessel kernel for chiral matrix models in the microscopic limit

TL;DR

This paper proves that the correlation functions of chiral complex matrix models universally follow the Bessel kernel in the microscopic limit, revealing a fundamental aspect of eigenvalue behavior near the origin.

Contribution

It demonstrates the universality of the Bessel kernel for chiral matrix models in the microscopic limit, using the reduction of difference equations to differential equations.

Findings

Correlation functions follow the Bessel kernel universally in the microscopic limit.

The difference equation reduces to a Bessel differential equation in the limit.

The proof applies to chiral complex matrix models near the eigenvalue origin.

Abstract

We prove the universality of correlation functions of chiral complex matrix models in the microscopic limit (N->\infty, z->0, N z=fixed) which magnifies the crossover region around the origin of the eigenvalue distribution. The proof exploits the fact that the three-term difference equation for orthogonal polynomials reduces into a universal second-order differential (Bessel) equation in the microscopic limit.