Deformations of the Tracy-Widom distribution

TL;DR

This paper investigates how the Tracy-Widom distribution, key in random matrix theory, transforms under two models involving disorder and eigenvalue removal, revealing continuous transitions to Gaussian and Weibull distributions.

Contribution

It introduces two models showing how the Tracy-Widom distribution deforms, expanding understanding of eigenvalue behavior under disorder and eigenvalue removal.

Findings

Disorder in Gaussian ensembles causes TW to transition to Gaussian distribution.

Removing eigenvalues leads to a transition from TW to Weibull distribution.

The formalism of Fredholm determinants extends to these new models.

Abstract

In random matrix theory (RMT), the Tracy-Widom (TW) distribution describes the behavior of the largest eigenvalue. We consider here two models in which TW undergoes transformations. In the first one disorder is introduced in the Gaussian ensembles by superimposing an external source of randomness. A competition between TW and a normal (Gaussian) distribution results, depending on the spreading of the disorder. The second model consists in removing at random a fraction of (correlated) eigenvalues of a random matrix. The usual formalism of Fredholm determinants extends naturally. A continuous transition from TW to the Weilbull distribution, characteristc of extreme values of an uncorrelated sequence, is obtained.