First passage percolation in hostile environment is not monotone

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 (in a "delayed" manner) from a random set of vertices distributed according to Bernoulli percolation of parameter $\mu\in (0,1)$, and spreads at some fixed rate $\lambda>0$. In previous works (cf. [SS19, CS, FS]) it has been shown that when $\mu$ is small enough then there is a non-empty range of values for $\lambda$ such that the cluster eventually infected by $FPP_1$ can be infinite with positive probability. However the probability of this event is zero if $\mu$ is large enough. It might seem intuitive that the probability of obtaining an infinite $FPP_1$ cluster is a monotone function of $\mu$. In this work, we prove that, in general, this claim is false by constructing a graph for which one can find two values $0<\mu_1<\mu_2<1$ such that for all $\lambda$ small enough, if $\mu=\mu_1$ then the sought probability is zero, and if $\mu=\mu_2$ then this probability is bounded away from zero.