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 $FPP_\lambda$, spreading on a graph. $FPP_1$ starts at the origin and spreads at rate $1$, whereas $FPP_\lambda$ starts from a random set of \emph{inactive seeds} distributed as Bernoulli percolation of parameter $\mu\in (0,1)$. A seed of $FPP_\lambda$ gets activated when one of the two processes attempts to occupy its location, and from this moment onwards spreads at some fixed rate $\lambda>0$. In previous works~[17, 3, 7] it has been shown that when both $\mu$ or $\lambda$ are small enough, then $FPP_1$ \emph{survives} (i.e., it occupies an infinite set of vertices) with positive probability. It might seem intuitive that decreasing $\mu$ or $\lambda$ 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 $\mu$ or $\lambda$, implying the existence of multiple phase transitions. This behavior contrasts with other natural growth processes such as the $2$-type Richardson model.