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

TL;DR

This paper compares the size of percolation clusters in symmetric vertex sets, establishing stochastic domination results under symmetry conditions, with applications to various graph structures and models.

Contribution

It introduces a general stochastic domination result for cluster sizes in symmetric sets, extending to various random partition models and graph types.

Findings

Cluster size in V+ stochastically dominates that in V- under symmetry.

Results apply to Bernoulli percolation, site percolation, and the random cluster model.

Applicable to bunkbed, layered, lattice, and hypercube graphs.

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.