Asymptotics for the Covariance of the Airy_2 process

TL;DR

This paper derives higher-order asymptotic expansions for the Airy_2 process covariance, expressing it in terms of Tracy-Widom GUE distribution moments up to tenth order, extending previous asymptotic analyses.

Contribution

It provides a systematic method to represent asymptotic approximations of the Airy_2 process covariance as polynomials and integrals involving Painlevé II functions, up to tenth order.

Findings

Asymptotic expansion expressed as polynomials and integrals of Painlevé II functions.

Covariance approximated up to tenth order using Tracy-Widom GUE distribution moments.

Extended previous work by providing higher-order terms in the asymptotic expansion.

Abstract

In this paper we compute some of the higher order terms in the large-t asymptotic expansion of the Airy process two-point function, extending the previous work of Adler and van Moerbeke and Widom. We prove that it is possible to represent any order asymptotic approximation as a polynomial and integrals of the Hastings-McLeod Painlev\'e II function and its first derivative. Further, for up to tenth order we give this asymptotic approximation as a linear combination of the Tracy-Widom GUE density function f_2 and its derivatives. As a corollary to this, the asymptotic covariance is expressed up to tenth order in terms of the moments of the Tracy-Widom GUE distribution.