Gap Probabilities in Non-Hermitian Random Matrix Theory

TL;DR

This paper analyzes the probability of finding exactly k eigenvalues within a circle for various non-Hermitian random matrix ensembles, providing new explicit formulas and asymptotic expansions.

Contribution

It introduces new explicit Fredholm determinant and Pfaffian representations for gap probabilities in non-Hermitian ensembles, including non-Gaussian and Gaussian weights, and explores their asymptotic behavior.

Findings

Derived explicit Fredholm eigenvalues for chiral ensembles with Gaussian weights.

Established asymptotic expansion of the gap probability for large radius r and matrix size N.

Provided bounds and numerical evidence for gap probability behavior depending on zero eigenvalues.

Abstract

We compute the gap probability that a circle of radius r around the origin contains exactly k complex eigenvalues. Four different ensembles of random matrices are considered: the Ginibre ensembles and their chiral complex counterparts, with both complex (beta=2) or quaternion real (beta=4) matrix elements. For general non-Gaussian weights we give a Fredholm determinant or Pfaffian representation respectively, depending on the non-Hermiticity parameter. At maximal non-Hermiticity, that is for rotationally invariant weights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 by skipping every second factor, in contrast to the known relation for Hermitian ensembles. On additionally choosing Gaussian weights we give new explicit expressions for the Fredholm eigenvalues in the chiral case, in terms of Bessel-K and incomplete Bessel-I functions. This compares to known results for the Ginibre ensembles in terms of incomplete exponentials. Furthermore we present an asymptotic expansion of the logarithm of the gap probability for large argument r at large N in all four ensembles, up to including the third order linear term. We can provide strict upper and lower bounds and present numerical evidence for its conjectured values, depending on the number of exact zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at beta=2 exact results were previously derived by Forrester.