First passage percolation in hostile environment is not monotone

Elisabetta Candellero, Alexandre Stauffer

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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 $mu0,1)$, and spreads at some fixed rate $0$. In previous works (cf. [SS19, CS, FS]) it has been shown that when $ is small enough then there is a non-empty range of values for $ such that the cluster eventually infected by $FPP_1$ can be infinite with positive probability. However the probability of this event is zero if $ is large enough. It might seem intuitive that the probability of obtaining an infinite $FPP_1$ cluster is a monotone function of $. In this work, we prove that, in general, this claim is false by constructing a graph for which one can find two values $0
Original languageEnglish
JournalElectronic Journal Of Probability
Publication statusAccepted/In press - 14 May 2024


  • math.PR


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