Supremum of the Airy2 process minus a parabola on a half line

Abstract

Let $\aip(t)$ be the Airy${}_2$ process. We show that the random variable [\sup_{t\leqα}\{aip(t)-t^2}+\min{0,α}^2] has the same distribution as the one-point marginal of the Airy${}_{2\to 1}$ process at time $\alpha$. These marginals form a family of distributions crossing over from the GUE Tracy-Widom distribution $F_{\mathrm{GUE}}(x)$ for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE Tracy-Widom distribution $F_{\mathrm{GOE}}(4^{1/3}x)$ for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every $\alpha$ the distribution has the same right tail decay $e^{-(4/3)x^{3/2}}$.

To appear in Journal of Statistical Physics

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Cited by in corpus (8)

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• Multiplicative functionals on ensembles of non-intersecting paths
• Tails of the endpoint distribution of directed polymers
• Upper tail decay of KPZ models with Brownian initial conditions
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• KPZ fluctuations in finite volume