Macroscopic Hausdorff dimension of the level sets of the Airy processes

TL;DR

This paper investigates the macroscopic Hausdorff dimension of level sets of Airy processes, providing new bounds and estimates by leveraging inequalities and tail probability analysis for these stochastic processes.

Contribution

It introduces novel methods for analyzing the macroscopic Hausdorff dimension of Airy process level sets, extending previous techniques to these specific stochastic models.

Findings

Derived bounds for the Hausdorff dimension of Airy$_1$ level sets.

Established tail probability estimates for the maximum and minimum of Airy$_2$.

Extended the methodology for macroscopic dimension analysis of stochastic processes.

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.