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

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.