On the multipoint distribution formulas of the parabolic Airy process

TL;DR

This paper proves the equivalence of two different formulas for the multipoint distribution of the parabolic Airy process, providing new formulas and a generalization of Andreief's identity, enhancing understanding of KPZ-related stochastic processes.

Contribution

It offers a direct proof that a recent multipoint distribution formula matches the classical extended Airy kernel formula for the parabolic Airy process, including new formulas and a generalized identity.

Findings

Confirmed the equivalence of two multipoint distribution formulas

Derived new formulas for the parabolic Airy process

Generalized Andreief's identity

Abstract

The parabolic Airy process is the Airy$_2$ process minus a parabola, initially defined by its finite-dimensional distributions, which are given by a Fredholm determinant formula with the extended Airy kernel. This process is also the one-time spatial marginal of the KPZ fixed point with the narrow wedge initial condition. There are two formulas for the space-time multipoint distribution of the KPZ fixed point with the narrow wedge initial condition obtained by arXiv:1906.01053 and arXiv:1907.09876. Especially, the equal-time case of arXiv:1907.09876 gives a different formula of the multipoint distribution of the parabolic Airy process. In this paper, we present a direct proof that this formula matches the one with the extended Airy kernel. Some byproducts in the proof include several new formulas for the parabolic Airy process, and a generalization of the Andreief's identity.