Cauchy-Laguerre two-matrix model and the Meijer-G random point field

TL;DR

This paper develops explicit formulas for a new random point field called the Meijer-G field, arising from the Cauchy-Laguerre two-matrix model, and explores its universality and limiting behaviors.

Contribution

It introduces the Meijer-G random field as a new universality class in random matrix theory, with explicit formulas and convergence results.

Findings

Existence of a scaling limit for eigenvalues near the origin.

Introduction of the Meijer-G random field as a new universality class.

Convergence of the Meijer-G field to the Bessel field in a limit.

Abstract

We apply the general theory of Cauchy biorthogonal polynomials developed previously by the authors, to the case associated with Laguerre measures. In particular, we obtain explicit formulae in terms of Meijer-G functions for all key objects relevant to the study of the corresponding biorthogonal polynomials and the Cauchy two-matrix model associated with them. The central theorem we prove is that a scaling limit of the correlation functions for eigenvalues near the origin exists, and is given by a new determinantal two--level random point field, the "Meijer-G random field". We conjecture that this random point field leads to a novel universality class of random fields parametrized by exponents of Laguerre weights. We express the joint distributions of the smallest eigenvalues in terms of suitable Fredholm determinants and evaluate them numerically. We also show that in a suitable limit, the Meijer-G random field converges to the Bessel random field and hence the behavior of the eigenvalues of one of the two matrices converges to the one of the Laguerre ensemble.