Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation

TL;DR

This paper studies the regularity and geometric properties of the Airy line ensemble, using its Brownian Gibbs property to establish bounds and decay rates relevant to Brownian last passage percolation and related models.

Contribution

It introduces a new technique for bounding probabilities of events in systems with the Brownian Gibbs property and determines a key exponent for disjoint near geodesics in Brownian last passage percolation.

Findings

Finiteness of a superpolynomial moment bound on Radon-Nikodym derivatives.

Decay rate of probability for existence of disjoint near geodesics.

Comparison between the Airy line ensemble's curves and Brownian bridges.

Abstract

The Airy line ensemble is a positive-integer indexed system of random continuous curves whose finite dimensional distributions are given by the multi-line Airy process. It is a natural object in the KPZ universality class: for example, its highest curve, the Airy$_2$ process, describes after the subtraction of a parabola the limiting law of the scaled energy of a geodesic running from the origin to a variable point on an anti-diagonal line in such problems as Poissonian last passage percolation. The ensemble of curves resulting from the Airy line ensemble after the subtraction of the same parabola enjoys a simple and explicit spatial Markov property, the Brownian Gibbs property. In this paper, we employ the Brownian Gibbs property to make a close comparison between the Airy line ensemble's curves after affine shift and Brownian bridge, proving the finiteness of a superpolynomially growing moment bound on Radon-Nikodym derivatives. We also determine the value of a natural exponent describing in Brownian last passage percolation the decay in probability for the existence of several near geodesics that are disjoint except for their common endpoints, where the notion of `near' refers to a small deficit in scaled geodesic energy, with the parameter specifying this nearness tending to zero. To prove both results, we introduce a technique that may be useful elsewhere for finding upper bounds on probabilities of events concerning random systems of curves enjoying the Brownian Gibbs property. Several results in this article play a fundamental role in a further study of Brownian last passage percolation in three companion papers, [Ham17a,b,c], in which geodesic coalescence and geodesic energy profiles are investigated in scaled coordinates.