Limit theorems for extrema of Airy processes

TL;DR

This paper proves limit theorems for the maxima and minima of Airy processes over growing intervals, identifying precise almost sure limits and constants, extending previous results with a new approach based on last passage percolation.

Contribution

It introduces a novel method for establishing limit theorems for Airy processes using last passage percolation convergence and sharp estimates, extending prior work on growth orders and constants.

Findings

Identified almost sure limits for maxima and minima of Airy processes.

Determined finite non-zero constants for the growth of extrema.

Extended previous results by providing new limit theorems and constants.

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.