Coexistence in competing first passage percolation with conversion

Thomas Finn, Alexandre Stauffer

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We introduce a two-type first passage percolation competition model on infinite connected graphs as follows. Type 1 spreads through the edges of the graph at rate 1 from a single distinguished site, while all other sites are initially vacant. Once a site is occupied by type 1, it converts to type 2 at rate ρ >0. Sites occupied by type 2 then spread at rate λ>0 through vacant sites and sites occupied by type 1, whereas type 1 can only spread through vacant sites. If the set of sites occupied by type 1 is nonempty at all times, we say type 1 survives. In the case of a regular d-ary tree for d ≥ 3, we show type 1 can survive when it is slower than type 2, provided ρ is small enough. This is in contrast to when the underlying graph is Zd , where for any ρ >0, type 1 dies out almost surely if λ > λ ' for some λ ' < 1.

Original languageEnglish
Pages (from-to)4459-4480
Number of pages22
JournalAnnals of Applied Probability
Issue number6
Publication statusPublished - 31 Dec 2022

Bibliographical note

Funding Information:
Funding. TF was supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1. AS was supported by EPSRC Fellowship EP/N004566/1.


  • coexistence
  • First passage percolation
  • random growth

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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