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

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 may be studied as a function of $z \in \mathbb{R}$. This profile is known to manifest a one-half power law, having $1/2-$-H\"older 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 poly-logarithmic correction. For a very broad class of initial data, the polymer weight profile has a modulus of continuity of the order of $x^{1/2} \big( \log x^{-1} \big)^{2/3}$, with a high degree of uniformity in the scaling parameter and the initial condition.