Supremum of the Airy_{2} Process Minus a Parabola on a Half Line
Abstract
Let {A}_{2}(t) be the Airy_{2} process. We show that the random variable sup_{t≤α} bigl\{{A}_2(t)t^2 bigr\}+min\{0,α \}^2 has the same distribution as the onepoint marginal of the Airy_{2→1} process at time α. These marginals form a family of distributions crossing over from the GUE TracyWidom distribution F _{GUE}( x) for the Gaussian Unitary Ensemble of random matrices, to a rescaled version of the GOE TracyWidom distribution F _{GOE}(4^{1/3} x) for the Gaussian Orthogonal Ensemble. Furthermore, we show that for every α the distribution has the same right tail decay e^{4/3 x^{3/2} }.
 Publication:

Journal of Statistical Physics
 Pub Date:
 February 2013
 DOI:
 10.1007/s1095501206334
 arXiv:
 arXiv:1111.2565
 Bibcode:
 2013JSP...150..442Q
 Keywords:

 Airy processes;
 Last passage percolation;
 KPZ universality;
 Mathematics  Probability;
 Mathematical Physics
 EPrint:
 To appear in Journal of Statistical Physics