First Passage Percolation Has Sublinear Distance Variance
Let 0 < a < b < ∞, and for each edge e of Zdlet ωe=a or ωe=b, each with probability 1/2, independently. This induces a random metric distωon the vertices of Zd, called first passage percolation. We prove that for d > 1, the distance distω(0, v) from the origin to a vertex v, |v| > 2,...
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| Veröffentlicht in: | The Annals of probability 2003-10, Vol.31 (4), p.1970-1978 |
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| Hauptverfasser: | , , |
| Format: | Artikel |
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| container_end_page | 1978 |
|---|---|
| container_issue | 4 |
| container_start_page | 1970 |
| container_title | The Annals of probability |
| container_volume | 31 |
| creator | Benjamini, Itai Kalai, Gil Schramm, Oded |
| description | Let 0 < a < b < ∞, and for each edge e of Zdlet ωe=a or ωe=b, each with probability 1/2, independently. This induces a random metric distωon the vertices of Zd, called first passage percolation. We prove that for d > 1, the distance distω(0, v) from the origin to a vertex v, |v| > 2, has variance bounded by C|v|/log |v|, where C = C (a, b, d) is a constant which may only depend on a, b and d. Some related variants are also discussed. |
| doi_str_mv | 10.1214/aop/1068646373 |
| format | Article |
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| ispartof | The Annals of probability, 2003-10, Vol.31 (4), p.1970-1978 |
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| language | eng |
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| source | Project Euclid; JSTOR; EZB Electronic Journals Library |
| subjects | 28A35 60B15 60E15 60K35 Coordinate systems discrete cube discrete harmonic analysis discrete isoperimetric inequalities Distribution theory Exact sciences and technology Harmonic analysis Hypercontractive influences Mathematical analysis Mathematical induction Mathematical inequalities Mathematics Measure and integration Probability and statistics Probability theory and stochastic processes Probability theory on algebraic and topological structures random metrics Research universities Sciences and techniques of general use Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications) Statistical variance Vertices |
| title | First Passage Percolation Has Sublinear Distance Variance |
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