Heavy repulsion of clusters in Bernoulli percolation
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911157100478464 |
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| author | Bell, Sasha Chu, Tasmin Rodgers, Owen Terlov, Grigory Tserunyan, Anush |
| author_facet | Bell, Sasha Chu, Tasmin Rodgers, Owen Terlov, Grigory Tserunyan, Anush |
| contents | We study Bernoulli$(p)$ percolation on (non)unimodular quasi-transitive graphs and prove that, almost surely, for any two heavy clusters $C$ and $C'$, the set of vertices in $C$ within distance one of $C'$ is light, i.e. it has finite total weight. This is a significant step towards resolving a longstanding question posed by Häggström, Peres, and Schonmann, and a generalization of a theorem of Timár, who proved the same result in the unimodular setting. Our proof adapts Timár's approach but requires developing weighted analogues of several classical unimodular results. This presents nontrivial challenges, since in a nonunimodular graph a subtree with infinitely many ends may be hyperfinite or even light. To overcome this, we employ newly developed machinery from the theory of measure-class-preserving equivalence relations and graphs. In particular, we establish a weighted generalization of a theorem of Benjamini, Lyons, and Schramm on the existence of an invariant random subgraph with positive weighted Cheeger constant, a result of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_10631 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Heavy repulsion of clusters in Bernoulli percolation Bell, Sasha Chu, Tasmin Rodgers, Owen Terlov, Grigory Tserunyan, Anush Probability Dynamical Systems We study Bernoulli$(p)$ percolation on (non)unimodular quasi-transitive graphs and prove that, almost surely, for any two heavy clusters $C$ and $C'$, the set of vertices in $C$ within distance one of $C'$ is light, i.e. it has finite total weight. This is a significant step towards resolving a longstanding question posed by Häggström, Peres, and Schonmann, and a generalization of a theorem of Timár, who proved the same result in the unimodular setting. Our proof adapts Timár's approach but requires developing weighted analogues of several classical unimodular results. This presents nontrivial challenges, since in a nonunimodular graph a subtree with infinitely many ends may be hyperfinite or even light. To overcome this, we employ newly developed machinery from the theory of measure-class-preserving equivalence relations and graphs. In particular, we establish a weighted generalization of a theorem of Benjamini, Lyons, and Schramm on the existence of an invariant random subgraph with positive weighted Cheeger constant, a result of independent interest. |
| title | Heavy repulsion of clusters in Bernoulli percolation |
| topic | Probability Dynamical Systems |
| url | https://arxiv.org/abs/2509.10631 |