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Non-ergodicity for the noisy majority vote process on trees

Jian Ding, Fenglin Huang

TL;DR

The paper analyzes the noisy majority vote process on infinite regular trees $\mathbb T_d$ with $d\ge 3$ and proves non-ergodicity (existence of multiple equilibrium measures) for small noise $\epsilon$, extending prior results that were restricted to $d\ge 5$. The authors develop an induction framework based on backward update histories and cluster decompositions, bounding the probability that a set of odd vertices becomes $-1$ by a product over non-adjacent odd clusters with penalties for trifurcations. Central to the argument are new combinatorial lemmas that bound the enumeration of induced configurations (via notions like single/double trifurcations and odd/even clusters) and an extension to general trees through a minus-biased vote process that is dominated by the original dynamics. The results thus establish non-ergodicity for all $d\ge 3$ and small $\epsilon$, highlighting how disorder persists and prevents unique long-term behavior on trees even under modest stochastic perturbations. These insights advance the understanding of ergodicity-breaking phenomena in noisy spin systems on hierarchical graphs and extend Burris–Gray-type results beyond the previously known regimes.

Abstract

We consider the noisy majority vote process on infinite regular trees with degree $d\geq 3$, and we prove the non-ergodicity, i.e., there exist multiple equilibrium measures. Our work extends a result of Bramson and Gray (2021) for $d\geq 5$.

Non-ergodicity for the noisy majority vote process on trees

TL;DR

The paper analyzes the noisy majority vote process on infinite regular trees with and proves non-ergodicity (existence of multiple equilibrium measures) for small noise , extending prior results that were restricted to . The authors develop an induction framework based on backward update histories and cluster decompositions, bounding the probability that a set of odd vertices becomes by a product over non-adjacent odd clusters with penalties for trifurcations. Central to the argument are new combinatorial lemmas that bound the enumeration of induced configurations (via notions like single/double trifurcations and odd/even clusters) and an extension to general trees through a minus-biased vote process that is dominated by the original dynamics. The results thus establish non-ergodicity for all and small , highlighting how disorder persists and prevents unique long-term behavior on trees even under modest stochastic perturbations. These insights advance the understanding of ergodicity-breaking phenomena in noisy spin systems on hierarchical graphs and extend Burris–Gray-type results beyond the previously known regimes.

Abstract

We consider the noisy majority vote process on infinite regular trees with degree , and we prove the non-ergodicity, i.e., there exist multiple equilibrium measures. Our work extends a result of Bramson and Gray (2021) for .

Paper Structure

This paper contains 16 sections, 31 theorems, 75 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Noisy majority vote process
  3. Related works
  4. A word on proof strategy for $\mathbb{T}_3$
  5. The extension to $d\ge 3$
  6. Proof of Theorem \ref{['thm: minus main higher dimension']}: the induction procedure
  7. Notations
  8. Induction hypothesis via update history
  9. The case of a single minus cluster
  10. Enumeration of $B$
  11. The case of multiple minus clusters
  12. Proof of Lemmas \ref{['lem: enumeration induced relation']} and \ref{['lem: enumeration lemma final']}
  13. Proof of Lemma \ref{['lem: enumeration induced relation']}
  14. Proof of Lemma \ref{['lem: enumeration lemma final']}: A simple case
  15. Proof of Lemma \ref{['lem: enumeration lemma final']}: The general case
  16. ...and 1 more sections

Key Result

Theorem 1.1

For any $d\ge 3$, the noisy majority vote process on $\mathbb T_d$ is not ergodic for small enough $\epsilon>0$.

Figures (9)

  • Figure 1: Illustrations of single and double trifurcations. Red point: a type 3 single trifurcation. Blue point: a type 3 double trifurcation. Green points: vertices in $A$.
  • Figure 2: An illustration for induced relation. Green points: the odd cluster $C$. Blue points: the even cluster $D$ induced by $C$.
  • Figure 3: An illustration of double trifurcations in $B$. Green points: a subset of $A$. Blue points: the even cluster $B$ induced by $A$.
  • Figure 4: An illustration of ${G}_A$ and $G_A^*$. The vertices with red boundaries: odd vertices. The vertices with black boundaries: even vertices. The numbers near the vertices are their orders in $G_A^*$. The green lines: the edges between a branching vertex and its far neighbor.
  • Figure 5: An illustration of $\tilde{G}_A$. The blue lines: the edges that are not in $G_A^*$.
  • ...and 4 more figures

Theorems & Definitions (87)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • ...and 77 more