A phase transition for competition interfaces
We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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2007
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| _version_ | 1826305359752462336 |
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| author | Ferrari, P Martin, J Pimentel, L |
| author_facet | Ferrari, P Martin, J Pimentel, L |
| author_sort | Ferrari, P |
| collection | OXFORD |
| description | We study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle $\theta$ of the cone: for $\theta\geq180^{\circ}$, the direction is deterministic, while for $\theta<180^{\circ}$, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times. |
| first_indexed | 2024-03-07T06:31:42Z |
| format | Journal article |
| id | oxford-uuid:f6322710-98a9-4a54-be9c-e2c3f038e597 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:31:42Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:f6322710-98a9-4a54-be9c-e2c3f038e5972022-03-27T12:33:22ZA phase transition for competition interfacesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:f6322710-98a9-4a54-be9c-e2c3f038e597EnglishSymplectic Elements at Oxford2007Ferrari, PMartin, JPimentel, LWe study the competition interface between two growing clusters in a growth model associated to last-passage percolation. When the initial unoccupied set is approximately a cone, we show that this interface has an asymptotic direction with probability 1. The behavior of this direction depends on the angle $\theta$ of the cone: for $\theta\geq180^{\circ}$, the direction is deterministic, while for $\theta<180^{\circ}$, it is random, and its distribution can be given explicitly in certain cases. We also obtain partial results on the fluctuations of the interface around its asymptotic direction. The evolution of the competition interface in the growth model can be mapped onto the path of a second-class particle in the totally asymmetric simple exclusion process; from the existence of the limiting direction for the interface, we obtain a new and rather natural proof of the strong law of large numbers (with perhaps a random limit) for the position of the second-class particle at large times. |
| spellingShingle | Ferrari, P Martin, J Pimentel, L A phase transition for competition interfaces |
| title | A phase transition for competition interfaces |
| title_full | A phase transition for competition interfaces |
| title_fullStr | A phase transition for competition interfaces |
| title_full_unstemmed | A phase transition for competition interfaces |
| title_short | A phase transition for competition interfaces |
| title_sort | phase transition for competition interfaces |
| work_keys_str_mv | AT ferrarip aphasetransitionforcompetitioninterfaces AT martinj aphasetransitionforcompetitioninterfaces AT pimentell aphasetransitionforcompetitioninterfaces AT ferrarip phasetransitionforcompetitioninterfaces AT martinj phasetransitionforcompetitioninterfaces AT pimentell phasetransitionforcompetitioninterfaces |