Large deformations of the Tracy-Widom distribution I. Non-oscillatory asymptotics

TL;DR

This paper rigorously analyzes the transition of the largest eigenvalue distribution from Tracy-Widom to Weibull statistics in random matrix ensembles by computing asymptotics of thinned distributions, revealing new insights into eigenvalue fluctuations.

Contribution

It provides the first rigorous computation of the non-oscillatory left-tail asymptotics for thinned Tracy-Widom distributions across different symmetry classes, advancing understanding of the transition between Tracy-Widom and Weibull laws.

Findings

Derived asymptotics for thinned GOE, GUE, GSE distributions

Established the transition behavior as thinning parameter varies

Connected results to Painlevé II equation solutions

Abstract

We analyze the left-tail asymptotics of deformed Tracy-Widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability $1-\gamma\in[0,1]$. As $\gamma$ varies, a transition from Tracy-Widom statistics ($\gamma=1$) to classical Weibull statistics ($\gamma=0$) was observed in the physics literature by Bohigas, de Carvalho, and Pato \cite{BohigasCP:2009}. We provide a description of this transition by rigorously computing the leading-order left-tail asymptotics of the thinned GOE, GUE and GSE Tracy-Widom distributions. In this paper, we obtain the asymptotic behavior in the non-oscillatory region with $\gamma\in[0,1)$ fixed (for the GOE, GUE, and GSE distributions) and $\gamma\uparrow 1$ at a controlled rate (for the GUE distribution). This is the first step in an ongoing program to completely describe the transition between Tracy-Widom and Weibull statistics. As a corollary to our results, we obtain a new total-integral formula involving the Ablowitz-Segur solution to the second Painlev\'e equation.