Quantitative homogenization of the disordered $\nabla \phi$ model

TL;DR

This paper establishes quantitative convergence rates for the surface tension and field norms in the disordered bla  model, using new tools from optimal transport and stochastic homogenization.

Contribution

It introduces a novel approach that avoids Helffer-Sjstrand representation, leveraging variational methods, displacement convexity, and stochastic homogenization for the bla  model.

Findings

Proves quantitative convergence rates for finite-volume surface tension.

Provides quantitative estimates for the bla  field with affine boundary conditions.

Develops a new toolbox for analyzing disordered bla  models.

Abstract

We study the $\nabla \phi$ model with uniformly convex Hamiltonian $\mathcal{H} (\phi) := \sum V(\nabla \phi)$ and prove a quantitative rate of convergence for the finite-volume surface tension as well as a quantitative rate estimate for the $L^2$-norm for the field subject to affine boundary condition. One of our motivations is to develop a new toolbox for studying this problem that does not rely on the Helffer-Sj\"ostrand representation. Instead, we make use of the variational formulation of the partition function, the notion of displacement convexity from the theory of optimal transport, and the recently developed theory of quantitative stochastic homogenization.