On the association of the Airy$_1$ process

Fei Pu

Published 2023-11-19Version 1

We first show that the Airy$_1$ process is associated using the association property of solution to stochastic heat equation and convergence of the KPZ equation to the KPZ fixed point. Then we apply Newman's inequality to establish the ergodicity and central limit theorem for the Airy$_1$ process. Combining with the asymptotic behavior of the GOE Tracy-Widom distribution, we derive a Poisson limit theorem for the Airy$_1$ process and give some estimate on the asymptotic behavior of the maximum of the the Airy$_1$ process over an interval. Analogous results for the Airy$_2$ process are also presented.