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Imry-Ma phenomenon for the hard-core model on $\mathbb{Z}^{2}$

Irene Ayuso Ventura, Leandro Chiarini, Tyler Helmuth, Ellen Powell

TL;DR

This work demonstrates an Imry-Ma-type phenomenon for the two-dimensional hard-core model: arbitrarily weak disorder from site percolation destroys crystal-like order and yields a unique infinite-volume Gibbs measure for $G_p$ with $p<1$, extending to random site-dependent activities $\boldsymbol{\lambda}=\{\lambda X_v\}$. The authors adapt the Aizenman–Wehr framework to a constrained lattice gas by exploiting spatial symmetries rather than internal spin symmetries, and establish Gaussian domination of the random-field contribution while tightly bounding boundary effects. The result aligns with the intuition from random-field Ising models and shows disorder can preclude phase transitions in 2D particle systems, with potential extensions to higher dimensions and other disorder types. The methods provide a blueprint for analyzing crystallization under weak, spatially structured randomness and may inform algorithmic and theoretical questions in disordered bipartite graphs.

Abstract

The \emph{Imry-Ma phenomenon} refers to the dramatic effect that disorder can have on first-order phase transitions for two-dimensional spin systems. The most famous example is the absence of a phase transition for the two-dimensional random-field Ising model. This paper establishes that a similar phenomena takes place for the hard-core model, a discrete model of crystallization: arbitrarily weak disorder prevents the formation of a crystal. Our proof of this behaviour is an adaptation of the Aizenman-Wehr argument for the Imry-Ma phenomenon, with the use of internal (spin space) symmetries for spin systems being replaced by the use spatial symmetries.

Imry-Ma phenomenon for the hard-core model on $\mathbb{Z}^{2}$

TL;DR

This work demonstrates an Imry-Ma-type phenomenon for the two-dimensional hard-core model: arbitrarily weak disorder from site percolation destroys crystal-like order and yields a unique infinite-volume Gibbs measure for with , extending to random site-dependent activities . The authors adapt the Aizenman–Wehr framework to a constrained lattice gas by exploiting spatial symmetries rather than internal spin symmetries, and establish Gaussian domination of the random-field contribution while tightly bounding boundary effects. The result aligns with the intuition from random-field Ising models and shows disorder can preclude phase transitions in 2D particle systems, with potential extensions to higher dimensions and other disorder types. The methods provide a blueprint for analyzing crystallization under weak, spatially structured randomness and may inform algorithmic and theoretical questions in disordered bipartite graphs.

Abstract

The \emph{Imry-Ma phenomenon} refers to the dramatic effect that disorder can have on first-order phase transitions for two-dimensional spin systems. The most famous example is the absence of a phase transition for the two-dimensional random-field Ising model. This paper establishes that a similar phenomena takes place for the hard-core model, a discrete model of crystallization: arbitrarily weak disorder prevents the formation of a crystal. Our proof of this behaviour is an adaptation of the Aizenman-Wehr argument for the Imry-Ma phenomenon, with the use of internal (spin space) symmetries for spin systems being replaced by the use spatial symmetries.
Paper Structure (11 sections, 6 theorems, 81 equations, 3 figures)

This paper contains 11 sections, 6 theorems, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Future Directions and Broader Context
  3. Acknowledgements
  4. Preliminaries
  5. Notation and basic definitions
  6. Infinite-volume Gibbs measures on bipartite graphs
  7. Proof of the main theorem
  8. Proofs of the main lemmas
  9. Occupation probabilities by differentiation
  10. Controlling the effect of boundary conditions
  11. Gaussian Domination

Key Result

theorem 1.1

Fix $\lambda\geq 0$, $p\in [0,1)$, and let $G_{p}$ be the random subgraph of $\mathbb{Z}^{2}$ determined by Bernoulli site percolation on $\mathbb{Z}^{2}$. Then $\mathcal{G}_{\lambda}(G_{p})$ is almost surely a singleton set.

Figures (3)

  • Figure 1: The set of occupied vertices (filled circles) on the left is not independent: the two vertices contained in the red edge are adjacent. The set on the right is independent.
  • Figure 2: Illustration of ${\mathbf{x}}_{\Lambda_{2}^{c}}$ as defined in \ref{['eq:switch-off']}. The outer bold square delimits $\Lambda_5$, the inner one delimits $\Lambda_2$. Sites $v \in \Lambda_5 \setminus \Lambda_2$ (in grey) have activity $\lambda_v = \lambda \, x_v$, while $\lambda_v = \lambda$ for $v \in \Lambda_2$.
  • Figure 3: Two independent sets on $\Lambda_4$ with boundary conditions specified on $\Lambda_5$. On the left, the configuration has even boundary conditions (orange). On the right, the reflected configuration has odd boundary conditions (blue).

Theorems & Definitions (11)

  • theorem 1.1
  • theorem 1.2
  • theorem 2.1
  • lemma 3.1
  • remark 3.2: Important!
  • lemma 3.3
  • lemma 3.4
  • proof : Proof of \ref{['thm:main-full']} given \ref{['lem:dG', 'lem:bound-conditional', 'lem:gaussian-domination']}.
  • proof : Proof of Lemma \ref{['lem:dG']}
  • proof : Proof of \ref{['lem:bound-conditional']}
  • ...and 1 more