Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)

Thomas Richthammer

TL;DR

The paper tackles monotonicity questions for Bernoulli percolation by comparing the size of the infection cluster $\mathcal{C}_o$ within two symmetric vertex-sets $V_+$ and $V_-$ under strong symmetry. It develops a general framework based on group actions and symmetry (Sections 2–4) and proves a core orbit-counting identity (Section 3) that yields a stochastic domination $|\mathcal{C}_-| \preceq |\mathcal{C}_+|$ conditioned on finiteness, with extensions to random partitions (Section 4). This approach is then specialized to percolation settings, showing that $\mathcal{C}_o\cap V_-$ is dominated by $\mathcal{C}_o\cap V_+$ in various symmetric graph classes (Section 5), and yielding corollaries about layer- and product-graph structures, including the $2$D square lattice and hypercube. Overall, the work provides a robust, model-agnostic method for monotonicity results in percolation and related random-partition processes, expanding understanding of how symmetry governs infection spread.

Abstract

For Bernoulli percolation on a given graph $G = (V,E)$ we consider the cluster of some fixed vertex $o \in V$. We aim at comparing the number of vertices of this cluster in the set $V_+$ and in the set $V_-$, where $V_+,V_- \subset V$ have the same size. Intuitively, if $V_-$ is further away from $o$ than $V_+$, it should contain fewer vertices of the cluster. We prove such a result in terms of stochastic domination, provided that $o \in V_+$, and $V_+,V_-$ satisfy some strong symmetry conditions, and we give applications of this result in case $G$ is a bunkbed graph, a layered graph, the 2D square lattice or a hypercube graph. Our result only relies on general probabilistic techniques and a combinatorial result on group actions, and thus extends to fairly general random partitions, e.g. as induced by Bernoulli site percolation or the random cluster model.

Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)

TL;DR

The paper tackles monotonicity questions for Bernoulli percolation by comparing the size of the infection cluster within two symmetric vertex-sets and under strong symmetry. It develops a general framework based on group actions and symmetry (Sections 2–4) and proves a core orbit-counting identity (Section 3) that yields a stochastic domination conditioned on finiteness, with extensions to random partitions (Section 4). This approach is then specialized to percolation settings, showing that is dominated by in various symmetric graph classes (Section 5), and yielding corollaries about layer- and product-graph structures, including the D square lattice and hypercube. Overall, the work provides a robust, model-agnostic method for monotonicity results in percolation and related random-partition processes, expanding understanding of how symmetry governs infection spread.

Abstract

For Bernoulli percolation on a given graph we consider the cluster of some fixed vertex . We aim at comparing the number of vertices of this cluster in the set and in the set , where have the same size. Intuitively, if is further away from than , it should contain fewer vertices of the cluster. We prove such a result in terms of stochastic domination, provided that , and satisfy some strong symmetry conditions, and we give applications of this result in case is a bunkbed graph, a layered graph, the 2D square lattice or a hypercube graph. Our result only relies on general probabilistic techniques and a combinatorial result on group actions, and thus extends to fairly general random partitions, e.g. as induced by Bernoulli site percolation or the random cluster model.
Paper Structure (5 sections, 12 theorems, 37 equations, 1 figure)

This paper contains 5 sections, 12 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Group actions on pairs of sets
  4. Random partitions
  5. Independent bond percolation

Key Result

Theorem 1

We consider Bernoulli percolation with parameter $p \in (0,1)$ on a connected, locally finite graph $G = (V,E)$. Let $\mathcal{C} := \mathcal{C}_o$ denote the cluster of a fixed vertex $o \in V$. Let $V_+,V_- \subset V$ be two disjoint sets of vertices and let $\Gamma$ be a subgroup of $Aut(G)$ such Let $\mathcal{C}_+ := \mathcal{C} \cap V_+$, $\mathcal{C}_- := \mathcal{C} \cap V_-$ and $\mathcal{

Figures (1)

  • Figure 1: 20 classes of possible choices of $V_+,V_-$ in Corollary \ref{['Cor:Z2']}, where every class is illustrated by a typical representative. Here $\mathbbm{Z}^2$ is embedded into $\mathbbm{R}^2$ in the canonical way. The vertices of $V_+$ are marked in black, the vertices of $V_-$ are marked in white. The origin $o$ is in one of the vertices of $V_+$. For classes 3-20 the pattern extends periodically. For some of the classes the set $V_\pm = V_+ \cup V_-$ is the same: for classes 1-2 the set consists of 4 vertices, for 3-4 all vertices are on a single line, for 5-6 and for 7-11 all vertices are on two parallel lines, for 12,13-15 and 16-20 the vertices are on infinitely many parallel lines. Case (a) of the corollary corresponds to the classes 4,5, and case (b) to 12,14.

Theorems & Definitions (17)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • Definition 1
  • Definition 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Theorem 2
  • ...and 7 more