First passage percolation in hostile environment is not monotone

Abstract

We study a natural growth process with competition, modeled by two first passage percolation processes, ${FPP}_1$ and ${FPP}_{\mathit{\lambda }}$, spreading on a graph. ${FPP}_1$ starts at the origin and spreads at rate 1, whereas ${FPP}_{\mathit{\lambda }}$ starts from a random set of inactive seeds distributed as Bernoulli percolation of parameter $\mathrm{\mu }\in (0,1)$. A seed of ${FPP}_{\mathit{\lambda }}$ gets activated when one of the two processes attempts to occupy its location, and from this moment onwards spreads at some fixed rate $\mathit{\lambda }>0$. In previous works [17, 3, 7] it has been shown that when both μ or λ are small enough, then ${FPP}_1$ survives (i.e., it occupies an infinite set of vertices) with positive probability. It might seem intuitive that decreasing μ or λ is beneficial to ${FPP}_1$. However, we prove that, in general, this is indeed false by constructing a graph for which the probability that ${FPP}_1$ survives is not a monotone function of μ or λ, implying the existence of multiple phase transitions. This behavior contrasts with other natural growth processes such as the 2-type Richardson model.