The right tail exponent of the Tracy-Widom-beta distribution

TL;DR

This paper derives the asymptotic tail behavior of the Tracy-Widom-beta distribution, revealing how the probability of large deviations decays exponentially with a power-law correction, using the stochastic Airy operator.

Contribution

It provides a precise asymptotic formula for the tail of the Tracy-Widom-beta distribution as the variable tends to infinity, employing the stochastic Airy operator representation.

Findings

Tail probability decays as a^(-3/4 beta+o(1)) exp(-2/3 beta a^(3/2))

Asymptotic behavior characterized for large a

Uses stochastic Airy operator to derive results

Abstract

The Tracy-Widom beta distribution is the large dimensional limit of the top eigenvalue of beta random matrix ensembles. We use the stochastic Airy operator representation to show that as a tends to infinity the tail of the Tracy Widom distribution satisfies P(TW_beta > a) = a^(-3/4 beta+o(1)) exp(-2/3 beta a^(3/2)).