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 FPP₁ and FPP_λ, spreading with rates 1 and λ > 0 respectively, on a graph G. FPP₁ starts from a single vertex at the origin o, while the initial configuration of FPP_λ consists of infinitely many seeds distributed according to a product of Bernoulli measures of parameter μ > 0 on V (G) \ {o}. FPP₁ starts spreading from time 0, while each seed of FPP_λ only starts spreading after it has been reached by either FPP₁ or FPP_λ. 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 λ > 0 there is a μ0 = μ₀(G, λ) > 0 such that for all μ ∈ (0, μ₀) the two processes coexist with positive probability. This is the first non-trivial instance where coexistence is established for this model.