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

Article number  85 
Pages (fromto)  142 
Journal  Electronic Journal of Probability 
Volume  29 
Early online date  17 Jun 2024 
DOIs  
Publication status  Epub ahead of print  17 Jun 2024 
Bibliographical note
.Keywords
 first passage percolation in hostile environment
 FPP
 monotonicity
ASJC Scopus subject areas
 Statistics and Probability
 Statistics, Probability and Uncertainty
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 1 Finished

Early Career Fellowship  Mathematical Analysis of Strongly Correlated Processes on Discrete Dynamic Structures
Stauffer, A.
Engineering and Physical Sciences Research Council
1/04/16 → 30/09/22
Project: Research council