Coexistence in two-type first-passage percolation models

By Olivier Garet, R. Marchand. 2003

78 citations recorded by the source corpus.

We study the problem of coexistence in a two-type competition model governed by first-passage percolation on Zd or on the infinite cluster in Bernoulli percolation. We prove for a large class of ergodic stationary passage times that for distinct points x, y ∈ Zd, there is a strictly positive probability that {z ∈ Zd; d(y, z) d(x, z)} are both infinite sets. We also show that there is a strictly positive probability that the graph of time-minimizing path from the origin in first-passage percolation has at least two topological ends. This generalizes results obtained by Haggstrom and Pemantle for independent exponential times on the square lattice.

Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/f01759c99890152cdbb5096148076ae6d6a0b2a6

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In Condensed Matter Physics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Coexistence for Richardson type competing spatial growth models (2004, background): “Garet and Marchand have given a difierent proof of Theorem 2 [ 3 ].”
Geodesics in first passage percolation (2005, unknown): “35 (1995) 683–692], Garet and Marchand [Ann.”
50 years of first passage percolation (2015, result): “Shortly thereafter, Garet and Marchand [79] and Hoffman [101] independently extended these results to prove coexistence of 2 species for a broad class of translation-invariant measures, including some non-i.i.d. ones.”