Supremum of the Airy2 Process Minus a Parabola on a Half Line

Abstract

Let A2(t) be the Airy2 process. We show that the random variable sup t≤α A2(t) −t 2 +min{0,α}2 has the same distribution as the one-point marginal of the Airy2→1 process at time α. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution FGUE(x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution FGOE(41/3x) for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every α the distribution has the same right tail decay e −43 x3/2 .