Convergence to the thermodynamic limit for random-field random surfaces

TL;DR

This paper proves the convergence of finite-volume distributions to infinite-volume Gibbs measures for random surfaces with quenched disorder, establishing dimension-dependent results and convergence rates.

Contribution

It establishes the convergence of surface and gradient distributions to Gibbs measures in high dimensions, with explicit convergence rates, advancing understanding of disordered surface models.

Findings

Gradient distribution converges in dimensions d≥4.

Surface distribution converges in dimensions d≥5.

Power-law rate of convergence in Wasserstein distance.

Abstract

We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient. In this work we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions $d\geq4$ while the distribution of the surface itself converges in dimensions $d\geq 5$. Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by Cotar and K\"ulske