On Asymptotics for the Airy Process

The Airy process t→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 Pr(A(0)≤s ₁, 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→∞, 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.