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Fixation for $\mathcal{U}$-Ising and $\mathcal{U}$-voter dynamics with frozen vertices

Laure Marêché

TL;DR

The paper analyzes fixation in the $\\mathcal{U}$-Ising and $\\mathcal{U}$-voter dynamics with frozen vertices on $\\mathbb{Z}^d$, extending classic results for the zero-temperature Ising model and voter model. It develops a two-pronged proof strategy: for non-supercritical update families it constructs good droplets that guarantee fixation inside regions, while for many supercritical families it exploits overlap-based propagation of $+$ to control flippers; plus a 1D analysis. In 2D, under condition $(\\mathcal{C})$, fixation to $+$ holds when $\\rho^- =0$, and for small $\\rho^-$, non-fixating components become finite, with a universality dichotomy: update families with two disjoint rules yield infinitely many flippers, whereas those without disjoint rules yield no flippers. The work thereby extends Damron et al.’s results to the $\\mathcal{U}$-Ising and $\\mathcal{U}$-voter dynamics, providing a dimensional bound and a universality classification across a broad class of update families. It also outlines limitations in higher dimensions and sketches novel arguments needed for the supercritical regime without disjoint update rules.

Abstract

The zero-temperature stochastic Ising model is a special case of the famous stochastic Ising model of statistical mechanics, and the voter model is another classical model in this field. In both models, each vertex of the graph $\mathbb{Z}^d$ can have one of two states, and can change state to match the state of its neighbors. In 2017, Morris proposed generalizations of these models, the $\mathcal{U}$-Ising and $\mathcal{U}$-voter dynamics, in which a vertex can change state to match the state of certain subsets of vertices near it. These generalizations were inspired by similar generalizations in the related model of bootstrap percolation, where Balister, Bollobás, Duminil-Copin, Morris, Przykucki, Smith and Uzzell were able to establish a very impressive universality classification of the generalized models. However, there have been very few results on the $\mathcal{U}$-Ising and $\mathcal{U}$-voter dynamics. The only one is due to Blanquicett in 2021, who obtained a few encouraging advances on the important question of fixation, which is only partially solved for the zero-temperature stochastic Ising model: will all vertices eventually settle on a given state or will they oscillate forever between the two states? In this work, we tackle a question which was solved for the zero-temperature stochastic Ising model by Damron, Eckner, Kogan, Newman and Sidoravicius in 2015: fixation when a fraction of the vertices of $\mathbb{Z}^d$ are frozen in one of the states. For $d=1$ and $2$, in most cases we prove that if all frozen vertices are in the same state, all vertices eventually settle at this state. Moreover, if vertices can be frozen in both states but the proportion of vertices frozen in the second state is small enough, we were able to establish a universality classification identifying the models in which all vertices settle in a given state.

Fixation for $\mathcal{U}$-Ising and $\mathcal{U}$-voter dynamics with frozen vertices

TL;DR

The paper analyzes fixation in the -Ising and -voter dynamics with frozen vertices on , extending classic results for the zero-temperature Ising model and voter model. It develops a two-pronged proof strategy: for non-supercritical update families it constructs good droplets that guarantee fixation inside regions, while for many supercritical families it exploits overlap-based propagation of to control flippers; plus a 1D analysis. In 2D, under condition , fixation to holds when , and for small , non-fixating components become finite, with a universality dichotomy: update families with two disjoint rules yield infinitely many flippers, whereas those without disjoint rules yield no flippers. The work thereby extends Damron et al.’s results to the -Ising and -voter dynamics, providing a dimensional bound and a universality classification across a broad class of update families. It also outlines limitations in higher dimensions and sketches novel arguments needed for the supercritical regime without disjoint update rules.

Abstract

The zero-temperature stochastic Ising model is a special case of the famous stochastic Ising model of statistical mechanics, and the voter model is another classical model in this field. In both models, each vertex of the graph can have one of two states, and can change state to match the state of its neighbors. In 2017, Morris proposed generalizations of these models, the -Ising and -voter dynamics, in which a vertex can change state to match the state of certain subsets of vertices near it. These generalizations were inspired by similar generalizations in the related model of bootstrap percolation, where Balister, Bollobás, Duminil-Copin, Morris, Przykucki, Smith and Uzzell were able to establish a very impressive universality classification of the generalized models. However, there have been very few results on the -Ising and -voter dynamics. The only one is due to Blanquicett in 2021, who obtained a few encouraging advances on the important question of fixation, which is only partially solved for the zero-temperature stochastic Ising model: will all vertices eventually settle on a given state or will they oscillate forever between the two states? In this work, we tackle a question which was solved for the zero-temperature stochastic Ising model by Damron, Eckner, Kogan, Newman and Sidoravicius in 2015: fixation when a fraction of the vertices of are frozen in one of the states. For and , in most cases we prove that if all frozen vertices are in the same state, all vertices eventually settle at this state. Moreover, if vertices can be frozen in both states but the proportion of vertices frozen in the second state is small enough, we were able to establish a universality classification identifying the models in which all vertices settle in a given state.
Paper Structure (13 sections, 10 theorems, 2 equations, 3 figures)

This paper contains 13 sections, 10 theorems, 2 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and results
  3. One-dimensional case
  4. Two-dimensional case
  5. Outline of the proofs
  6. Notation
  7. Non-supercritical update families: good droplets
  8. Easily fixating regions: good blocks
  9. Fixation at $+$: proof of Theorems \ref{['thm_fixation']} and \ref{['thm_connected_components']}
  10. Flippers: proof of Theorem \ref{['thm_flippers']}
  11. First case: $\mathcal{U}$ contains two disjoint update rules $X$ and $X'$.
  12. Second case: $\mathcal{U}$ contains no disjoint update rules.
  13. One-dimensional case: proof of Theorems \ref{['thm_fixation_dim1']} and \ref{['thm_flippers_dim1']}

Key Result

Theorem 3

If $\mathcal{U}$ is a supercritical one-dimensional update family, in the $\mathcal{U}$-Ising and the $\mathcal{U}$-voter dynamics with frozen vertices, if $0 < \rho^+ <1$ and $\rho^-=0$, for any choice of initial distribution $\mu$, almost surely all sites fixate at $+$.

Figures (3)

  • Figure 1: A two-dimensional supercritical update family with no disjoint update rules: $\mathcal{U}=\{\{(-1,0),(-1,1)\},\{(-1,0),(-1,-1)\}\}$. The bullets represent the sites contained in the update rules.
  • Figure 2: Droplets $D(a)$ and $D'(a)$ in the case $N=3$.
  • Figure 3: A representation of $C_{1,2}(a) = (\bar{\mathds{H}}_{-u_1}(a)\setminus\mathds{H}_{-u_1}(a-r)) \cap (\bar{\mathds{H}}_{-u_{2}}(a)\setminus\mathds{H}_{-u_{2}}(a-r))$, here the "bottom left corner of $D(a)$". $C_{1,2}(a)$ is the shape with the solid thick outline, and $D(a)$ the shape with the dashed thick outline. $\bar{\mathds{H}}_{-u_1}(a)$ and $\mathds{H}_{-u_1}(a-r)$ are represented with hatchings: $\mathds{H}_{-u_1}(a-r)$ is the region with dense hatchings, and $\bar{\mathds{H}}_{-u_1}(a)$ contains both this region and the region with the spaced hatchings. Similarly, $\mathds{H}_{-u_2}(a-r)$ is the region in dark gray and $\bar{\mathds{H}}_{-u_2}(a)$ contains both this region and the region in lighter gray.

Theorems & Definitions (33)

  • Definition 1
  • Definition 2
  • Theorem 3
  • Theorem 4
  • Definition 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • Theorem 10
  • ...and 23 more