{"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. "}],"citation":{"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.","ama":"Richthammer T. Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions). Published online 2022.","short":"T. Richthammer, (2022).","ieee":"T. Richthammer, “Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions).” 2022.","apa":"Richthammer, T. (2022). <i>Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)</i>.","mla":"Richthammer, Thomas. <i>Comparing the Number of Infected Vertices in Two Symmetric Sets for Bernoulli Percolation (and Other Random Partitions)</i>. 2022."},"type":"preprint","date_created":"2026-02-18T12:13:09Z","date_updated":"2026-02-18T12:13:21Z","title":"Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)","year":"2022","status":"public","author":[{"id":"62054","last_name":"Richthammer","first_name":"Thomas","full_name":"Richthammer, Thomas"}],"user_id":"62054","_id":"64214","language":[{"iso":"eng"}]}