---
_id: '64214'
abstract:
- lang: eng
  text: '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. '
author:
- first_name: Thomas
  full_name: Richthammer, Thomas
  id: '62054'
  last_name: Richthammer
citation:
  ama: Richthammer T. Comparing the number of infected vertices in two symmetric sets
    for Bernoulli percolation (and other random partitions). Published online 2022.
  apa: Richthammer, T. (2022). <i>Comparing the number of infected vertices in two
    symmetric sets for Bernoulli percolation (and other random partitions)</i>.
  bibtex: '@article{Richthammer_2022, title={Comparing the number of infected vertices
    in two symmetric sets for Bernoulli percolation (and other random partitions)},
    author={Richthammer, Thomas}, year={2022} }'
  chicago: Richthammer, Thomas. “Comparing the Number of Infected Vertices in Two
    Symmetric Sets for Bernoulli Percolation (and Other Random Partitions),” 2022.
  ieee: T. Richthammer, “Comparing the number of infected vertices in two symmetric
    sets for Bernoulli percolation (and other random partitions).” 2022.
  mla: Richthammer, Thomas. <i>Comparing the Number of Infected Vertices in Two Symmetric
    Sets for Bernoulli Percolation (and Other Random Partitions)</i>. 2022.
  short: T. Richthammer, (2022).
date_created: 2026-02-18T12:13:09Z
date_updated: 2026-02-18T12:13:21Z
language:
- iso: eng
status: public
title: Comparing the number of infected vertices in two symmetric sets for Bernoulli
  percolation (and other random partitions)
type: preprint
user_id: '62054'
year: '2022'
...