On the exponent governing the correlation decay of the Airy$_1$ process

TL;DR

This paper investigates the decay rate of the covariance of the Airy_1 process, revealing a super-exponential decay with a leading order term proportional to u^3, using probabilistic and integrable probability techniques.

Contribution

It establishes the super-exponential decay rate of the Airy_1 process covariance and determines the leading order exponent, connecting it to last passage percolation geometry.

Findings

Covariance decays as e^{-(4/3+o(1))u^3} for large u

New results on point-to-line geodesics in exponential last passage percolation

Fredholm determinant and FKG inequality used for bounds

Abstract

We study the decay of the covariance of the Airy$_1$ process, $\mathcal{A}_1$, a stationary stochastic process on $\mathbb{R}$ that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that $\textrm{Cov}(\mathcal{A}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$ as $u\to\infty$. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $\mathcal{A}_1$ to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the Airy$_1$ process together with the FKG inequality.