The Riemann-Hilbert approach to the generating function of the higher order Airy point processes

TL;DR

This paper establishes a Tracy-Widom type formula for the generating function of occupancy numbers in higher order Airy point processes, linking it to a new vector-valued Painlevé II hierarchy and its Lax pair.

Contribution

It introduces a novel Tracy-Widom type formula for higher order Airy processes and defines a new vector-valued Painlevé II hierarchy with its Lax pair.

Findings

Derived a Tracy-Widom type formula for the generating function

Connected the formula to a new Painlevé II hierarchy

Established the Lax pair for the hierarchy

Abstract

We prove a Tracy-Widom type formula for the generating function of occupancy numbers on several disjoint intervals of the higher order Airy point processes. The formula is related to a new vector-valued Painlev\'e II hierarchy we define, together with its Lax pair.