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A phase transition and critical phenomenon for the two-dimensional random field Ising model

Jian Ding, Fenglin Huang, Aoteng Xia

Abstract

We study the random field Ising model in a two-dimensional box with side length $N$ where the external field is given by independent normal variables with mean $0$ and variance $ε^2$. Our primary result is the following phase transition at $T = T_c$: for $ε\ll N^{-7/8}$ the boundary influence (i.e., the difference between the spin averages at the center of the box with the plus and the minus boundary conditions) decays as $N^{-1/8}$ and thus the disorder essentially has no effect on the boundary influence; for $ε\gg N^{-7/8}$, the boundary influence decays as $N^{-\frac{1}{8}}e^{-Θ(ε^{8/7}\, N)}$ (i.e., the disorder contributes a factor of $e^{-Θ(ε^{8/7}\, N)}$ to the decay rate). For a natural notion of the correlation length, i.e., the minimal size of the box where the boundary influence shrinks by a factor of $2$ from that with no external field, we also prove the following: as $ε\downarrow 0$ the correlation length transits from $Θ(ε^{-8/7})$ at $T_c$ to $e^{Θ(ε^{-4/3}\,\,)}$ for $T < T_c$.

A phase transition and critical phenomenon for the two-dimensional random field Ising model

Abstract

We study the random field Ising model in a two-dimensional box with side length where the external field is given by independent normal variables with mean and variance . Our primary result is the following phase transition at : for the boundary influence (i.e., the difference between the spin averages at the center of the box with the plus and the minus boundary conditions) decays as and thus the disorder essentially has no effect on the boundary influence; for , the boundary influence decays as (i.e., the disorder contributes a factor of to the decay rate). For a natural notion of the correlation length, i.e., the minimal size of the box where the boundary influence shrinks by a factor of from that with no external field, we also prove the following: as the correlation length transits from at to for .
Paper Structure (33 sections, 43 theorems, 269 equations, 3 figures)

This paper contains 33 sections, 43 theorems, 269 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Previous works
  3. Proof ingredients
  4. Preliminaries
  5. Notations
  6. FK-Ising model
  7. A continuous extension of the Ising model
  8. Disagreement percolation
  9. Basic properties of the model(s)
  10. FKG for pre-disagreements
  11. CBC for pre-disagreements
  12. Outline of the proof
  13. Proof of Theorem \ref{['thm-critical-temperature-small-perturbation']}
  14. Proof of Theorem \ref{['thm-critical-temperature']}
  15. Upper bound
  16. ...and 18 more sections

Key Result

Theorem 1.1

Fix $d=2$. For any constant $c_{1}>0$, there exists a constant $c_{2}=c_{2}(c_{1})>0$ such that for $\epsilon\geq c_{1}N^{-\frac{7}{8}}$, Furthermore, there exist constants $c_{3},c_{4}>0$ such that for $\epsilon\geq c_{-1}N^{-\frac{7}{8}}$ the following holds with $\mathbb P$-probability at least $1-c_{-2}^{-1}\exp(-c_{-2}(\epsilon^{\frac{8}{7}}N)^{\frac{1}{10}})$:

Figures (3)

  • Figure 1: Outmost agreed boundary around $\mathcal{R}_2$
  • Figure 3: Good box. The red curves are open paths in $\omega|_{\mathsf Q}$.
  • Figure 4: An illustration for $\mathcal{R}_T, \mathcal{R}_B, \mathcal{R}_L, \mathcal{R}_R$ and $\mathcal{R}'$.

Theorems & Definitions (118)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Remark 2.4
  • Proposition 2.5
  • Proposition 2.6
  • ...and 108 more