Continuum Statistics of the Airy2 Process

Abstract

We develop an exact determinantal formula for the probability that the Airy2 process is bounded by a function g on a finite interval. As an application, we provide a direct proof that sup(A2(x) − x2) 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 (Commun. Math. Phys. 242(1–2):277–329, 2003) observation that the GOE result follows from this polymer interpretation and exact results within that field. In a companion paper (Moreno Flores et al. in Commun. Math. Phys. 2012) these continuum statistics are used to compute the distribution of the endpoint of directed polymers.