TY - JOUR

T1 - First passage percolation in hostile environment is not monotone

AU - Candellero, Elisabetta

AU - Stauffer, Alexandre

N1 - 36 pages, 6 figures

PY - 2024/5/14

Y1 - 2024/5/14

N2 - 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

AB - 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

KW - math.PR

M3 - Article

SN - 1083-6489

JO - Electronic Journal Of Probability

JF - Electronic Journal Of Probability

ER -