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Aizenman-Wehr argument for a class of disordered gradient models

Simon Buchholz, Codina Cotar

Abstract

We consider random gradient fields with disorder where the interaction potential $V_e$ on an edge $e$ can be expressed as $e^{-V_e(s)} = \int ρ(\mathrm{d}κ)\, e^{-κξ_e} e^{-\frac{κs^2}{2}}$. Here $ρ$ denotes a measure with compact support in $(0,\infty)$ and $ξ_e\in\mathbb{R}$ 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.

Aizenman-Wehr argument for a class of disordered gradient models

Abstract

We consider random gradient fields with disorder where the interaction potential on an edge can be expressed as . Here denotes a measure with compact support in and a nontrivial edge dependent disorder. We show that in dimension 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.
Paper Structure (6 sections, 19 theorems, 121 equations)

This paper contains 6 sections, 19 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Gradient Gibbs measures with disorder
  3. The disordered random conductance model
  4. Aizenman-Wehr argument for random conductance model
  5. Relation between gradient measures and random conductance model
  6. Acknowledgments

Key Result

Theorem 2.6

Let $d=2$. Assume that ${\mathbb{E}}(e^{t|\xi_e|})<\infty$ for all $t\in {\mathbb{R}}$ and assume that the distribution of $\xi_e$ is not concentrated on a single point. Then there exists for $V^\xi$ as above a unique shift covariant gradient Gibbs measure $\xi\to \mu[\xi]$ such that the annealed me for some $\varepsilon>0$ and all $e\in \mathbf{E}({\mathbb{Z}}^2)$.

Theorems & Definitions (46)

  • Definition 2.1: Finite volume $\varphi$-Gibbs measure
  • Definition 2.2: $\varphi$-Gibbs measure on ${\mathbb{Z}}^d$
  • Definition 2.3: Finite volume $\nabla\varphi$-Gibbs measure
  • Definition 2.4: $\nabla\varphi$-Gibbs measure on $\mathbf{E}({\mathbb{Z}}^d)$
  • Definition 2.5: Shift covariant gradient Gibbs measure
  • Theorem 2.6
  • Corollary 2.7
  • proof
  • Theorem 3.1: Holley inequality
  • proof
  • ...and 36 more