On the average of the Airy process and its time reversal

TL;DR

This paper establishes a novel distributional identity linking the supremum of the average of the Airy process and its time reversal to the maximum of two independent GUE Tracy-Widom variables, using last passage percolation models.

Contribution

It introduces a new distributional equivalence involving the Airy process, its time reversal, and Tracy-Widom distributions, with a novel proof approach.

Findings

The supremum of the average of the Airy process and its time reversal minus a parabola follows a specific distribution.

The distribution is equivalent to the maximum of two independent GUE Tracy-Widom variables.

The proof employs a rotationally symmetric directed last passage percolation model.

Abstract

We show that the supremum of the average of the Airy process and its time reversal minus a parabola is distributed as the maximum of two independent GUE Tracy-Widom random variables. The proof is obtained by considering a directed last passage percolation model with a rotational symmetry in two different ways. We also review other known identities between the Airy process and the Tracy-Widom distributions.