Continuum statistics of the Airy2 process

TL;DR

This paper derives an exact formula for the probability bounds of the Airy2 process, linking it to GOE distributions and applications in directed polymer endpoint statistics.

Contribution

It introduces a determinantal formula for the Airy2 process bounds and connects it to GOE distributions and polymer endpoint analysis.

Findings

Exact determinantal formula for Airy2 process bounds

Distribution of supremum related to GOE random variable

Applications to directed polymer endpoint distribution

Abstract

We develop an exact determinantal formula for the probability that the Airy$_2$ process is bounded by a function $g$ on a finite interval. As an application, we provide a direct proof that $\sup(\aip(x)-x^2)$ is distributed as a GOE random variable. Both the continuum formula and the GOE result have applications in the study of the end point of an unconstrained directed polymer in a disordered environment. We explain Johansson's [Joh03] observation that the GOE result follows from this polymer interpretation and exact results within that field. In a companion paper [MQR11] these continuum statistics are used to compute the distribution of the endpoint of directed polymers.