Distribution of Schmidt-like eigenvalues for Gaussian Ensembles of the Random Matrix Theory

TL;DR

This paper studies the distribution of a measure of eigenvalue deviation in Gaussian random matrices, showing it converges to the Marčenko-Pastur law for large matrices and providing explicit formulas for the unitary case.

Contribution

It derives the asymptotic distribution of the Schmidt-like eigenvalue variable for Gaussian ensembles and provides exact formulas for the unitary case for any matrix size.

Findings

Distribution converges to Marčenko-Pastur law as matrix size grows

Explicit formulas for the eigenvalue deviation distribution in the unitary case

Distribution supported on [0,4] with a specific square-root form

Abstract

We analyze the form of the probability distribution function P_{n}^{(\beta)}(w) of the Schmidt-like random variable w = x_1^2/(\sum_{j=1}^n x^{2}_j/n), where x_j are the eigenvalues of a given n \times n \beta-Gaussian random matrix, \beta being the Dyson symmetry index. This variable, by definition, can be considered as a measure of how any individual eigenvalue deviates from the arithmetic mean value of all eigenvalues of a given random matrix, and its distribution is calculated with respect to the ensemble of such \beta-Gaussian random matrices. We show that in the asymptotic limit n \to \infty and for arbitrary \beta the distribution P_{n}^{(\beta)}(w) converges to the Mar\v{c}enko-Pastur form, i.e., is defined as P_{n}^{(\beta)}(w) \sim \sqrt{(4 - w)/w} for w \in [0,4] and equals zero outside of the support. Furthermore, for Gaussian unitary (\beta = 2) ensembles we present exact explicit expressions for P_{n}^{(\beta=2)}(w) which are valid for arbitrary n and analyze their behavior.