Coexistence of competing first passage percolation on hyperbolic graphs

TL;DR

This paper investigates the coexistence of two competing growth processes on hyperbolic graphs, demonstrating conditions under which both processes form infinite clusters, highlighting fundamental differences from Euclidean lattice behavior.

Contribution

It establishes the first non-trivial conditions for coexistence of competing first passage percolation processes on hyperbolic graphs, especially for non-amenable, vertex transitive cases.

Findings

Coexistence occurs with positive probability on hyperbolic graphs for small seed density.

FPP_λ produces an infinite cluster almost surely for any positive λ and μ.

Distinct behavior from Euclidean lattice models is demonstrated.

Abstract

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 $\text{FPP}_1$ and $\text{FPP}_\lambda$, spreading with rates $1$ and $\lambda>0$ respectively, on a graph $G$. $\text{FPP}_1$ starts from a single vertex at the origin $o$, while the initial configuration of $\text{FPP}_\lambda$ consists of infinitely many \emph{seeds} distributed according to a product of Bernoulli measures of parameter $\mu>0$ on $V(G)\setminus \{o\}$. $\text{FPP}_1$ starts spreading from time 0, while each seed of $\text{FPP}_\lambda$ only starts spreading after it has been reached by either $\text{FPP}_1$ or $\text{FPP}_\lambda$. 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 $\mu_0=\mu_0(G,\lambda)>0$ such that for all $\mu\in(0,\mu_0)$ 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 $\text{FPP}_\lambda$ produces an infinite cluster almost surely for any positive $\lambda,\mu$, establishing fundamental differences with the behavior of such processes on $\mathbb{Z}^d$.