Smallest Gaps Between Eigenvalues of Random Matrices With Complex Ginibre, Wishart and Universal Unitary Ensembles

TL;DR

This paper investigates the distribution of the smallest eigenvalue gaps in three types of random matrices, revealing Poissonian behavior and specific scaling laws for each ensemble.

Contribution

It provides the first detailed analysis of the limiting distribution of the smallest eigenvalue gaps for Ginibre, Wishart, and universal unitary ensembles, including explicit density functions.

Findings

Smallest gaps follow a Poissonian distribution.

Different ensembles have distinct typical gap lengths and densities.

Explicit formulas for the distribution of the k-th smallest gap.

Abstract

In this paper we study the limiting distribution of the $k$ smallest gaps between eigenvalues of three kinds of random matrices -- the Ginibre ensemble, the Wishart ensemble and the universal unitary ensemble. All of them follow a Poissonian ansatz. More precisely, for the Ginibre ensemble we have a global result in which the $k$-th smallest gap has typical length $n^{-3/4}$ with density $x^{4k-1}e^{-x^4}$ after normalization. For the Wishart and the universal unitary ensemble, it has typical length $n^{-4/3}$ and has density $x^{3k-1}e^{-x^3}$ after normalization.