Local behavior and hitting probabilities of the Airy process

Abstract

We obtain a formula for the n-dimensional distributions of the Airy process in terms of a Fredholm determinant on L2(R), as opposed to the standard formula which involves extended kernels, on L2({1, . . . , n} × R). The formula is analogous to an earlier formula of Prähofer and Spohn (J Stat Phys 108(5–6):1071– 1106, 2002) for the Airy2 process. Using this formula we are able to prove that the Airy process is Hölder continuous with exponent 1 2—and that it fluctuates locally like a Brownian motion.We also explain how the same methods can be used to obtain the analogous results for the Airy process. As a consequence of these two results, we derive a formula for the continuum statistics of the Airy1 process, analogous to that obtained in Corwin et al. (CommunMath Phys 2012, to appear) for the Airy process.