Large deviations of the top eigenvalue of large Cauchy random matrices

TL;DR

This paper analytically derives the probability distribution of the largest eigenvalue in large, heavy-tailed Cauchy random matrices across different symmetry classes, extending Tracy-Widom results to non-Gaussian ensembles.

Contribution

It provides the first exact analytical expressions for the large deviation tails and the central regime of the largest eigenvalue distribution in Cauchy ensembles for all symmetry classes.

Findings

Exact large deviation tails for the largest eigenvalue are computed.

The central distribution generalizes Tracy-Widom to heavy-tailed matrices.

Analytical results are validated by numerical simulations.

Abstract

We compute analytically the probability density function (pdf) of the largest eigenvalue $\lambda_{\max}$ in rotationally invariant Cauchy ensembles of $N\times N$ matrices. We consider unitary ($\beta = 2$), orthogonal ($\beta =1$) and symplectic ($\beta=4$) ensembles of such heavy-tailed random matrices. We show that a central non-Gaussian regime for $\lambda_{\max} \sim \mathcal{O}(N)$ is flanked by large deviation tails on both sides which we compute here exactly for any value of $\beta$. By matching these tails with the central regime, we obtain the exact leading asymptotic behaviors of the pdf in the central regime, which generalizes the Tracy-Widom distribution known for Gaussian ensembles, both at small and large arguments and for any $\beta$. Our analytical results are confirmed by numerical simulations.