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.)

• Speaker: Elisabetta Candellero (Warwick)
• Tuesday 08 May 2018, 14:00–15:00
• Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
• Series: Probability; organiser: Perla Sousi.