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Coexistence in two-type first-passage percolation models

Olivier Garet·Regine Marchand
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Abstract

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. Actually, we prove for a large class of ergodic stationary passage times that for distinct points $x,y\in\Zd$, there is a strictly positive probability that $\{z\in\Zd;d(y,z) d(x,z)\}$ 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 H{ä}ggstr{ö}m and Pemantle for independent exponential times on the square lattice.

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Olivier Garet, Regine Marchand. 2003-12-18. Coexistence in two-type first-passage percolation models. https://arxiv.org/abs/math/0312369

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