Aizenman-Wehr argument for a class of disordered gradient models

TL;DR

This paper proves that for a class of disordered gradient models in two dimensions, phase transitions do not occur, by extending the Aizenman-Wehr argument to these models using a connection to random conductance models.

Contribution

It extends the Aizenman-Wehr argument to disordered gradient models with specific potentials, showing the absence of phase transitions in 2D.

Findings

Unique ergodic disordered gradient Gibbs measure in 2D

No phase transitions persist in the disordered setting

Connection to random conductance models enables the proof

Abstract

We consider random gradient fields with disorder where the interaction potential $V_e$ on an edge $e$ can be expressed as $e^{-V_e(s)} = \int \rho(\mathrm{d}\kappa)\, e^{-\kappa \xi_e} e^{-\frac{\kappa s^2}{2}}$. Here $\rho$ denotes a measure with compact support in $(0,\infty)$ and $\xi_e\in\mathbb{R}$ a nontrivial edge dependent disorder. We show that in dimension $d=2$ there is a unique shift covariant disordered gradient Gibbs measure such that the annealed measure is ergodic and has zero tilt. This shows that the phase transitions known to occur for this class of potential do not persist to the disordered setting. The proof relies on the connection of the gradient Gibbs measures to a random conductance model with compact state space, to which the well known Aizenman-Wehr argument applies.