First passage percolation in hostile environment is not monotone

Elisabetta Candellero, Alexandre Stauffer

Research output: Contribution to journalArticlepeer-review


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 from a random set of inactive seeds distributed as Bernoulli percolation of parameter µ ∈ (0, 1). A seed of FPPλ 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 FPP1 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 FPP1. However, we prove that, in general, this is indeed false by constructing a graph for which the probability that FPP1 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 2-type Richardson model.

Original languageEnglish
Article number85
Pages (from-to)1-42
JournalElectronic Journal of Probability
Early online date17 Jun 2024
Publication statusE-pub ahead of print - 17 Jun 2024

Bibliographical note



  • first passage percolation in hostile environment
  • FPP
  • monotonicity

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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