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

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.

16 pages, 1 figure, new version: some references and discussion added, minor simplifications

Cited by in corpus (1)

• Monotonicity properties for Bernoulli percolation on layered graphs -- a Markov chain approach