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Probability Seminars

First passage percolation in hostile environment (on hyperbolic graphs)

With Elisabetta Candellero (Warwick)

First passage percolation in hostile environment (on hyperbolic graphs)

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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