The Tracy-Widom distribution is not infinitely divisible

TL;DR

This paper proves that the Tracy-Widom distribution, which describes the largest eigenvalue in certain random matrix ensembles, is not infinitely divisible, revealing new limitations in the distribution's properties.

Contribution

It establishes that the $eta$-Tracy-Widom distribution and the largest eigenvalues of GOE/GUE matrices are not infinitely divisible, a novel insight into their mathematical structure.

Findings

The $eta$-Tracy-Widom distribution is not infinitely divisible.

Largest eigenvalues of GOE/GUE matrices are not infinitely divisible.

Provides new understanding of the distribution's properties.

Abstract

The classical infinite divisibility of distributions related to eigenvalues of some random matrix ensembles is investigated. It is proved that the $\beta$-Tracy-Widom distribution, which is the limiting distribution of the largest eigenvalue of a $\beta$-Hermite ensemble, is not infinitely divisible. Furthermore, for each fixed $N \ge 2$ it is proved that the largest eigenvalue of a GOE/GUE random matrix is not infinitely divisible.