Existence of random gradient states

TL;DR

This paper investigates the existence of gradient Gibbs measures in disordered gradient models across different dimensions, establishing conditions under which such measures exist or do not, and analyzing surface tension properties.

Contribution

It proves the existence of shift-covariant gradient Gibbs measures in higher dimensions for models with zero-mean disorder and characterizes surface tension properties, extending prior nonexistence results.

Findings

Existence of gradient Gibbs measures in model A for d≥3 with zero-mean disorder.

Existence of gradient Gibbs measures in model B for all dimensions with zero-mean disorder.

Nonexistence of such measures in model A when disorder has nonzero mean in d≥3.

Abstract

We consider two versions of random gradient models. In model A the interface feels a bulk term of random fields while in model B the disorder enters through the potential acting on the gradients. It is well known that for gradient models without disorder there are no Gibbs measures in infinite-volume in dimension d=2, while there are "gradient Gibbs measures" describing an infinite-volume distribution for the gradients of the field, as was shown by Funaki and Spohn. Van Enter and K\"{u}lske proved that adding a disorder term as in model A prohibits the existence of such gradient Gibbs measures for general interaction potentials in $d=2$. In the present paper we prove the existence of shift-covariant gradient Gibbs measures with a given tilt $u\in \mathbb{R}^d$ for model A when $d\geq3$ and the disorder has mean zero, and for model B when $d\geq1$. When the disorder has nonzero mean in model A, there are no shift-covariant gradient Gibbs measures for $d\ge3$. We also prove similar results of existence/nonexistence of the surface tension for the two models and give the characteristic properties of the respective surface tensions.