Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions)
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917965124861952 |
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| author | Richthammer, Thomas |
| author_facet | Richthammer, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_12183 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions) Richthammer, Thomas Probability Combinatorics 60K35, 05E18 (Primary) 05A18 (Secondary) 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. |
| title | Comparing the number of infected vertices in two symmetric sets for Bernoulli percolation (and other random partitions) |
| topic | Probability Combinatorics 60K35, 05E18 (Primary) 05A18 (Secondary) |
| url | https://arxiv.org/abs/2204.12183 |