Ergodicity, CLT and asymptotic maximum of the Airy${}_1$ process

We first show that the Airy${}_1$ process is associated using the association property of the solution to the 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. Combined with the asymptotic behavior of the tail probability, we derive a Poisson limit theorem for the Airy${}_1$ process and give a precise estimate on the asymptotic behavior of the maximum of the Airy${}_1$ process over an interval. Analogous results for the Airy${}_2$ process are also presented.