Séminaire de Probabilités commun ICJ/UMPA
# Coexistence of competing first passage percolation on hyperbolic graphs

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Europe/Paris

Fokko du Cloux (ICJ, La Doua)
### Fokko du Cloux

#### ICJ, La Doua

Description

We consider two first-passage percolation processes FPP_1 and FPP_{\lambda}, spreading with rates 1 and \lambda > 0 respectively, on a non-amenable hyperbolic graph G with bounded degree. FPP_1 starts from a single source at the origin of G, while the initial con figuration of FPP_{\lambda} consists of countably many seeds distributed according to a product of iid Bernoulli random variables of parameter \mu > 0 on V (G)\{o}. Seeds start spreading FPP_{\lambda} after they are reached by either FPP_1 or FPP_{\lambda}. We show that for any such graph G, and any fixed value of \lambda > 0 there is a value \mu_0 = \mu_0(G,\lambda ) > 0 such that for all 0 < \mu < \mu_0 the two processes coexist with positive probability. This shows a fundamental difference with the behavior of such processes on Z^d. (Joint work with Alexandre Stauffer.)