Phase transition for the speed of the biased random walk on the supercritical percolation cluster
<p>We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least two, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias, such that, be...
| Main Authors: | , |
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| Format: | Journal article |
| Language: | English |
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Wiley
2013
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| _version_ | 1797071060515946496 |
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| author | Fribergh, A Hammond, A |
| author_facet | Fribergh, A Hammond, A |
| author_sort | Fribergh, A |
| collection | OXFORD |
| description | <p>We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least two, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the sub-ballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk.</p><p>A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one-dimensional; we prove such a dynamic renormalization statement in a much stronger form than was previously known.</p> |
| first_indexed | 2024-03-06T22:47:46Z |
| format | Journal article |
| id | oxford-uuid:5dc5bf1a-7e34-4c6b-afa8-8d861ee8962d |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:47:46Z |
| publishDate | 2013 |
| publisher | Wiley |
| record_format | dspace |
| spelling | oxford-uuid:5dc5bf1a-7e34-4c6b-afa8-8d861ee8962d2022-03-26T17:36:24ZPhase transition for the speed of the biased random walk on the supercritical percolation clusterJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:5dc5bf1a-7e34-4c6b-afa8-8d861ee8962dStatistical mechanics,structure of matter (mathematics)EnglishOxford University Research Archive - ValetWiley2013Fribergh, AHammond, A<p>We prove the sharpness of the phase transition for the speed in biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least two, and for any supercritical parameter p > p_c, we prove the existence of a critical strength for the bias, such that, below this value, the speed is positive, and, above the value, it is zero. We identify the value of the critical bias explicitly, and, in the sub-ballistic regime, we find the polynomial order of the distance moved by the particle. Each of these conclusions is obtained by investigating the geometry of the traps that are most effective at delaying the walk.</p><p>A key element in proving our results is to understand that, on large scales, the particle trajectory is essentially one-dimensional; we prove such a dynamic renormalization statement in a much stronger form than was previously known.</p> |
| spellingShingle | Statistical mechanics,structure of matter (mathematics) Fribergh, A Hammond, A Phase transition for the speed of the biased random walk on the supercritical percolation cluster |
| title | Phase transition for the speed of the biased random walk on the supercritical percolation cluster |
| title_full | Phase transition for the speed of the biased random walk on the supercritical percolation cluster |
| title_fullStr | Phase transition for the speed of the biased random walk on the supercritical percolation cluster |
| title_full_unstemmed | Phase transition for the speed of the biased random walk on the supercritical percolation cluster |
| title_short | Phase transition for the speed of the biased random walk on the supercritical percolation cluster |
| title_sort | phase transition for the speed of the biased random walk on the supercritical percolation cluster |
| topic | Statistical mechanics,structure of matter (mathematics) |
| work_keys_str_mv | AT fribergha phasetransitionforthespeedofthebiasedrandomwalkonthesupercriticalpercolationcluster AT hammonda phasetransitionforthespeedofthebiasedrandomwalkonthesupercriticalpercolationcluster |