Continuum statistics of the Airy2 process

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.