On Asymptotics for the Airy Process

Abstract

The Airy process A(t), introduced by Prähofer and Spohn, is the limiting stationary process for a polynuclear growth model. Adler and van Moerbeke found a PDE in the variables s_1, s_2, and t for the probability that A(0)<s_1 and A(t)<s_2. Using this they were able, assuming the truth of a certain conjecture and appropriate uniformity, to obtain the first few terms of an asymptotic expansion for this probability as t->infinity, with fixed s_1 and s_2. We shall show that the expansion can be obtained by using the Fredholm determinant representation for the probability. The main ingredients are formulas obtained by the author and C. A. Tracy in the derivation of the Painlevé II representation for the distribution function F_2 plus a few others obtained in the same way.