Oriented first passage percolation in the mean field limit, 2. The extremal process
| Autor: | Nicola Kistler, Adrien Schertzer, Marius A. Schmidt |
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| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
60J80 60G70 82B44 Euclidean space Mathematical analysis Probability (math.PR) First passage percolation Cox process Gumbel distribution Derrida’s REMs mean field approximation FOS: Mathematics Second moment method Limit (mathematics) Statistics Probability and Uncertainty Contraction principle First-hitting-time model Mathematics - Probability Mathematics |
| Zdroj: | Ann. Appl. Probab. 30, no. 2 (2020), 788-811 |
| DOI: | 10.48550/arxiv.1808.04598 |
| Popis: | This is the second, and last paper in which we address the behavior of oriented first passage percolation on the hypercube in the limit of large dimensions. We prove here that the extremal process converges to a Cox process with exponential intensity. This entails, in particular, that the first passage time converges weakly to a random shift of the Gumbel distribution. The random shift, which has an explicit, universal distribution related to modified Bessel functions of the second kind, is the sole manifestation of correlations ensuing from the geometry of Euclidean space in infinite dimensions. The proof combines the multiscale refinement of the second moment method with a conditional version of the Chen-Stein bounds, and a contraction principle. Comment: 26 pages, 2 figures |
| Databáze: | OpenAIRE |
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