The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview
Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain a...
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| Format: | Article |
| Language: | English |
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Discrete Mathematics & Theoretical Computer Science
2003-01-01
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| Series: | Discrete Mathematics & Theoretical Computer Science |
| Subjects: | |
| Online Access: | https://dmtcs.episciences.org/3340/pdf |
| _version_ | 1797270617926402048 |
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| author | Dayue Chen Yuval Peres |
| author_facet | Dayue Chen Yuval Peres |
| author_sort | Dayue Chen |
| collection | DOAJ |
| description | Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail. |
| first_indexed | 2024-04-25T02:07:07Z |
| format | Article |
| id | doaj.art-4413c909ac2d4b049816f2047793db25 |
| institution | Directory Open Access Journal |
| issn | 1365-8050 |
| language | English |
| last_indexed | 2024-04-25T02:07:07Z |
| publishDate | 2003-01-01 |
| publisher | Discrete Mathematics & Theoretical Computer Science |
| record_format | Article |
| series | Discrete Mathematics & Theoretical Computer Science |
| spelling | doaj.art-4413c909ac2d4b049816f2047793db252024-03-07T14:29:56ZengDiscrete Mathematics & Theoretical Computer ScienceDiscrete Mathematics & Theoretical Computer Science1365-80502003-01-01DMTCS Proceedings vol. AC,...Proceedings10.46298/dmtcs.33403340The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an OverviewDayue Chen0Yuval Peres1School of Mathematics and Physics [Beijing]Department of Statistics [Berkeley]Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph $G$, and the simple random walk on it, that are preserved by random perturbations. To address problems raised by those authors, we study simple random walk on the infinite percolation cluster in Cayley graphs of certain amenable groups known as "lamplighter groups''.We prove that zero speed for random walk on a lamplighter group implies zero speed for random walk on an infinite cluster, for any supercritical percolation parameter $p$. For $p$ large enough, we also establish the converse. We prove that if $G$ has a positive anchored expansion constant then so does every infinite cluster of independent percolation with parameter $p$ sufficiently close to 1; We also show that positivity of the anchored expansion constant is preserved under a random stretch if, and only if, the stretching law has an exponential tail.https://dmtcs.episciences.org/3340/pdfanchored expansion constant.cayley graphspercolationrandom walksspeed[info.info-ds] computer science [cs]/data structures and algorithms [cs.ds][info.info-dm] computer science [cs]/discrete mathematics [cs.dm][math.math-co] mathematics [math]/combinatorics [math.co][info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| spellingShingle | Dayue Chen Yuval Peres The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview Discrete Mathematics & Theoretical Computer Science anchored expansion constant. cayley graphs percolation random walks speed [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| title | The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview |
| title_full | The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview |
| title_fullStr | The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview |
| title_full_unstemmed | The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview |
| title_short | The Speed of Simple Random Walk and Anchored Expansion on Percolation Clusters: an Overview |
| title_sort | speed of simple random walk and anchored expansion on percolation clusters an overview |
| topic | anchored expansion constant. cayley graphs percolation random walks speed [info.info-ds] computer science [cs]/data structures and algorithms [cs.ds] [info.info-dm] computer science [cs]/discrete mathematics [cs.dm] [math.math-co] mathematics [math]/combinatorics [math.co] [info.info-cg] computer science [cs]/computational geometry [cs.cg] |
| url | https://dmtcs.episciences.org/3340/pdf |
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