Ordered random walks and the Airy line ensemble

TL;DR

This paper proves a universality property of the Airy line ensemble by showing that certain conditioned random walks converge to it under specific growth conditions, linking discrete models to continuous universality classes.

Contribution

It establishes a universality result for the Airy line ensemble from a broad class of random walks with exponential moments and log-concave densities, extending previous continuous models.

Findings

Top particles converge to the Airy line ensemble in a specific scaling limit.

Law of large numbers and fluctuations match those of non-intersecting Brownian motions.

Convergence holds when the number of walks grows slower than a power of the number of steps.

Abstract

The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of the Airy line ensemble. We study growing numbers of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. We consider a general class of increment distributions; a sufficient condition is the existence of an exponential moment and a log-concave density. We prove that the top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power (with a non-optimal exponent 3/50) of the expected number of random walk steps. Furthermore, in a similar regime we prove that the law of large numbers and fluctuations of linear statistics agree with non-intersecting Brownian motions.