Modulus of continuity of polymer weight profiles in Brownian last passage percolation

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, one endpoint of such polymers is fixed, say at $(0,0)\in \mathbb{R}^{2}$, and the other is varied horizontally, over $(z,1)$, $z\in \mathbb{R}$, so that the polymer weight profile is a function of $z\in \mathbb{R}$. This profile is known to manifest a one-half power law, having $1/2$-Hölder continuity. The polymer weight profile may be defined beginning from a much more general initial condition. In this article, we present a more general assertion of this one-half power law, as well as a bound on the polylogarithmic correction. The polymer weight profile admits a modulus of continuity of order $x^{1/2}(\log x^{-1})^{2/3}$, with a high degree of uniformity in the scaling parameter and over a very broad class of initial data.