Coexistence of competing first passage percolation on hyperbolic graphs

Elisabetta Candellero, Alexandre Stauffer

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We study a natural growth process with competition, which was recently introduced to analyze MDLA, a challenging model for the growth of an aggregate by diffusing particles. The growth process consists of two first-passage percolation processes FPP 1 and FPP λ, spreading with rates 1 and λ > 0 respectively, on a graph G. FPP 1 starts from a single vertex at the origin o, while the initial configuration of FPP λ consists of infinitely many seeds distributed according to a product of Bernoulli measures of parameter μ > 0 on V (G) \ {o}. FPP 1 starts spreading from time 0, while each seed of FPP λ only starts spreading after it has been reached by either FPP 1 or FPP λ. A fundamental question in this model, and in growth processes with competition in general, is whether the two processes coexist (i.e., both produce infinite clusters) with positive probability. We show that this is the case when G is vertex transitive, non-amenable and hyperbolic, in particular, for any λ > 0 there is a μ0 = μ 0(G, λ) > 0 such that for all μ ∈ (0, μ 0) the two processes coexist with positive probability. This is the first non-trivial instance where coexistence is established for this model. We also show that FPP λ produces an infinite cluster almost surely for any positive λ, μ, establishing fundamental differences with the behavior of such processes on Z d

Original languageEnglish
Pages (from-to)2128-2164
Number of pages37
JournalAnnales de l'Institut Henri Poincaré: Probabilités et Statistiques
Issue number4
Publication statusPublished - 30 Nov 2021


  • Coexistence
  • Competition
  • First passage percolation
  • First passage percolation in hostile environment
  • Hyperbolic graphs
  • Non-amenable graphs
  • Two-type Richardson model

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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