Limit theorems for extrema of Airy processes

We establish limit theorems for the maxima and minima of Airy₁ and Airy₂ processes (denoted by A₁(·) and A₂(·) respectively) over growing intervals. In particular, we identify the finite non-zero constants that are the almost sure limits of (log t)-2/3max0≤ s ≤ t Aᵢ(s) and (log t)-1/3min0≤ s ≤ t Aᵢ(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 A₁ and A₂ and the constant was identified for A₁. 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 A₁ and A₂, 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.