A patchwork quilt sewn from Brownian fabric: regularity of polymer weight profiles in Brownian last passage percolation

TL;DR

This paper demonstrates that in Brownian last passage percolation, polymer weight profiles from general initial conditions resemble Brownian bridges locally, with controlled regularity and integrability properties, extending previous narrow wedge results.

Contribution

It generalizes the known Brownian profile regularity from narrow wedge initial conditions to more general initial conditions, showing local Brownian-like behavior with uniform integrability.

Findings

Polymer weight profiles are locally Brownian in the narrow wedge case.

Profiles from general initial conditions resemble Brownian bridges on compact intervals.

The Radon-Nikodym derivatives of the profiles are in all L^p spaces for p in (1,3).

Abstract

In last passage percolation models lying in the KPZ universality class, the energy of long energy-maximizing paths may be studied as a function of the paths' pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at $(0,0) \in \mathbb{R}^2$, and the other is varied horizontally, over $(z,1)$, $z \in \mathbb{R}$, the polymer weight profile as a function of $z \in \mathbb{R}$ is locally Brownian; indeed, by Theorem $2.11$ and Proposition $2.5$ of [Ham16], the law of the profile is known to enjoy a very strong comparison to Brownian bridge on a given compact interval, with a Radon-Nikodym derivative in every $L^p$ space for $p \in (1,\infty)$, uniformly in the scaling parameter, provided that an affine adjustment is made to the weight profile before the comparison is made. In this article, we generalize this narrow wedge case and study polymer weight profiles begun from a very general initial condition. We prove that the profiles on a compact interval resemble Brownian bridge in a uniform sense: splitting the compact interval into a random but controlled number of patches, the profile in each patch after affine adjustment has a Radon-Nikodym derivative that lies in every $L^p$ space for $p \in (1,3)$. This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in [Ham17a] using techniques from [Ham16] and [Ham17b].