2021/08/24 by Thomas Finn, Finn, Thomas, Alexandre Stauffer +1
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2108.10559
openalex publication_date 2021/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 non-empty 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>1.