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Abstract
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 firstpassage 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, nonamenable 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 nontrivial instance where coexistence is established for this model.
Original language  English 

Pages (fromto)  21282164 
Number of pages  37 
Journal  Annales de l'Institut Henri Poincaré: Probabilités et Statistiques 
Volume  57 
Issue number  4 
DOIs  
Publication status  Published  30 Nov 2021 
Bibliographical note
Funding Information:This work started when E. Candellero was affiliated to the University of Warwick. E. Candellero acknowledges support from the project “Programma per Giovani Ricercatori Rita Levi Montalcini” awarded by the Italian Ministry of Education and support by “INdAM – GNAMPA Project 2019”. A. Stauffer acknowledges support from an EPSRC Early Career Fellowship.
Publisher Copyright:
© 2021 Institute of Mathematical Statistics. All rights reserved.
Keywords
 Coexistence
 Competition
 First passage percolation
 First passage percolation in hostile environment
 Hyperbolic graphs
 Nonamenable graphs
 Twotype Richardson model
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