Macroscopic Hausdorff dimension of the level sets of the Airy processes

TL;DR

This paper determines the macroscopic (large-scale) multifractal geometry of the level sets of the Airy$_1$ and Airy$_2$ processes. By leveraging the KKX17 macroscopic Hausdorff framework, association inequalities, and sharp tail bounds from exponential Last Passage Percolation, it shows that the upper-level sets have dimension ${\rm Dim_H}=1-\gamma^{3/2}$ and the lower-level sets have dimension ${\rm Dim_H}=1-\gamma^{3}$ for $\gamma\in(0,1)$, revealing a precise large-scale fractal structure of tall peaks and deep valleys. The results unify the behavior of both Airy processes at macroscopic scales and emphasize the multifractal nature of their extremal height fluctuations. The methods connect stochastic growth models in the KPZ class with rigorous fractal-geometry analysis via probabilistic tail estimates and association properties.

Abstract

We study the Macroscopic Hausdorff dimension of the upper and lower level sets of the Airy processes, following the general method developed in Khoshnevisan et al. \cite{KKX17}. For the Airy$_1$ process, the approach to macroscopic Hausdorff dimension of level sets hinges on some inequalities for its joint probabilities, while for the Airy$_2$ process, we make use of some quantitative estimates on the tail probabilities of its maximum and minimum over an interval.

Theorems & Definitions (10)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: KKX17
• Proposition 2.2: KKX17
• Theorem 2.3: BGZ21
• Theorem 2.4: BaB24
• proof : Proof of \ref{['eq:dim']}
• Lemma 3.1
• proof
• proof : Proof of \ref{['eq:dimL1']}: upper bound