First passage percolation in hostile environment is not monotone

We study a natural growth process with competition, modeled by two first passage percolation processes, FPP1 and FPPλ, spreading on a graph. FPP1 starts at the origin and spreads at rate 1, whereas FPPλ starts (in a “delayed” manner) from a random set of vertices distributed according to Bernoulli percolation of parameter μ ∈ (0, 1), and spreads at some fixed rate λ > 0. In previous works (cf. [SS19, CS, FS]) it has been shown that when μ is small enough then there is a nonempty range of values for λ such that the cluster eventually infected by FPP1 can be infinite with positive probability. However the probability of this event is zero if μ is large enough. It might seem intuitive that the probability of obtaining an infinite FPP1 cluster is a monotone function of μ. In this work, we prove that, in general, this claim is false by constructing a graph for which one can find two values 0 < μ1 < μ2 < 1 such that for all λ small enough, if μ = μ1 then the sought probability is zero, and if μ = μ2 then this probability is bounded away from zero.