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First passage percolation in hostile environment (on hyperbolic graphs)

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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 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.)

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