Limit theorems for extrema of Airy processes

Abstract

We establish limit theorems for the maxima and minima of Airy$_1$ and Airy$_2$ processes (denoted by $\mathcal{A}_1(\cdot)$ and $\mathcal{A}_2(\cdot)$ respectively) over growing intervals. In particular, we identify the finite non-zero constants that are the almost sure limits of $(\log t)^{-2/3}\max_{0\le s \le t} \mathcal{A}_{i}(s)$ and $(\log t)^{-1/3}\min_{0\le s \le t} \mathcal{A}_{i}(s)$ for $i=1,2$. This complements and extends the results of (Pu, 2023), where the question for the maxima was considered and the order of growth was identified for both $\mathcal{A}_1$ and $\mathcal{A}_2$ and the constant was identified for $\mathcal{A}_1$. Our approach is different from that of (Pu, 2023); instead of complicated formulae for multi-point distributions, we rely on the well-known convergence of passage time profiles in planar exponential last passage percolation started from different initial conditions to $\mathcal{A}_1$ and $\mathcal{A}_2$, together with the recently developed sharp one-point estimates in (Baslingker et al., 2024) for the point-to-point and point-to-line passage times in exponential LPP and a combination of old and new results on the geometry of the LPP landscape.