## 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.)

- 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.