Coexistence of competing first passage percolation on hyperbolic graphs

We study a natural growth process with competition, which was recently introduced to analyze MDLA, a challenging model for the growth of an aggregate by diffusing particles. The growth process consists of two first-passage percolation processes FPP 1 and FPPx, spreading with rates 1 and lambda > 0 respectively, on a graph G. FPP 1 starts from a single vertex at the origin o, while the initial configuration of FPPx consists of infinitely many seeds distributed according to a product of Bernoulli measures of parameter mu > 0 on V (G) \ {o}. FPP1 starts spreading from time 0, while each seed of FPP lambda only starts spreading after it has been reached by either FPP1 or FPP lambda. A fundamental question in this model, and in growth processes with competition in general, is whether the two processes coexist (i.e., both produce infinite clusters) with positive probability. We show that this is the case when G is vertex transitive, non- amenable and hyperbolic, in particular, for any lambda > 0 there is a mu(0) = mu(0) (G, > lambda) such that for all mu is an element of (0, mu(o)) the two processes coexist with positive probability. This is the first non-trivial instance where coexistence is established for this model. We also show that FPP lambda produces an infinite cluster almost surely for any positive lambda,mu, establishing fundamental differences with the behavior of such processes on Z(d).