Quantitative homogenization of the maximal action of curves in a Brownian potential

TL;DR

This paper proves a quantitative homogenization result for a variational problem involving a Brownian potential, showing that the maximal action scales logarithmically with the domain size with high probability.

Contribution

It sharpens previous qualitative results by providing uniform bounds on the action, establishing a precise logarithmic scaling law in a scale-invariant setting.

Findings

Maximal action scales as a ln L + O(1) with high probability.

Provides bounds for the action that are locally uniform in boundary conditions.

Refines earlier homogenization results by using pointwise bounds on the optimizer.

Abstract

Motivated by an optimal-matching problem (Leighton-Shor) and the random-field Ising model (Aizenman-Wehr, Ding-Wirth), we consider a variational problem for graphs in $1+1$ dimension maximizing an action that is the difference of a field term given by integrating white noise over the subgraph on the one hand, and the Dirichlet integral of the (continuum) height function $h=h(x)$ on the other hand. This problem is scale-invariant in law, and requires a small-scale cut-off which we implement by restricting to $h$ that are piecewise linear on intervals of size $1$ and vanish at $x=0,L$. We show that with overwhelming probability, the maximal action $A$ satisfies $A=a\ln L+O(1)$ for a deterministic constant $a\in(0,\infty)$. This can be considered as a homogenization result that is quantitative in an optimal way. The present result sharpens a recent qualitative homogenization result by the authors with C. Wagner; it does so by finding bounds for the action that are locally uniform in the boundary conditions. Like the earlier result, the present one relies on pointwise bounds on the optimizer, as provided by Dembin-Elboim-Hadas-Peled in a more general setting. In the earlier work, the small-scale cut-off also involved an explicit discretization of the field term, yielding a Brownian potential that is i.i.d. in $x\in\mathbb{Z}$; this had the benefit of allowing for a comparison argument, but is inconvenient for the coarse-graining used here.