On the joint distribution of the maximum and its position of the Airy2 process minus a parabola

TL;DR

This paper proves the equivalence of two recently derived formulas for the joint distribution of the maximum and its position in the Airy2 process minus a parabola, which models the endpoint distribution of a directed polymer.

Contribution

It provides a direct proof that two different formulas for the joint distribution are equivalent, linking Airy function-based and Painleve II-based representations.

Findings

Confirmed the equivalence of the two formulas for the joint distribution.

Established a connection between Airy function and Painleve II representations.

Enhanced understanding of the mathematical structure underlying the Airy2 process.

Abstract

The maximal point of the Airy2 process minus a parabola is believed to describe the scaling limit of the end-point of the directed polymer in a random medium, which was proved to be true for a few specific cases. Recently two different formulas for the joint distribution of the location and the height of this maximal point were obtained, one by Moreno Flores, Quastel and Remenik, and the other by Schehr. The first formula is given in terms of the Airy function and an associated operator, and the second formula is expressed in terms of the Lax pair equations of the Painleve II equation. We give a direct proof that these two formulas are the same.