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 $FPP_lambda$, spreading with rates $1$ and $lambda>0$ respectively, on a graph $G$. $FPP_1$ starts from a single vertex at the origin $o$, while the initial configuration of $FPP_lambda$ consists of infinitely many seeds distributed according to a product of Bernoulli measures of parameter $mu>0$ on $V(G)setminus {o}$. $FPP_1$ starts spreading from time 0, while each seed of $FPP_lambda$ only starts spreading after it has been reached by either $FPP_1$ or $FPP_lambda$. 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 $lambda>0$ there is a $mu_0=mu_0(G,lambda)>0$ such that for all $muin(0,mu_0)$ 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 $FPP_lambda$ produces an infinite cluster almost surely for any positive $lambda,mu$, establishing fundamental differences with the behavior of such processes on $mathbb{Z}^d$.

Candellero, E., DE OLIVEIRA STAUFFER, A. (2021). Coexistence of competing first passage percolation on hyperbolic graphs. ANNALES DE L'INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, in attesa di pubblicazione.

Coexistence of competing first passage percolation on hyperbolic graphs

Elisabetta Candellero
;
Alexandre Stauffer
2021-01-01

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 first-passage percolation processes $FPP_1$ and $FPP_lambda$, spreading with rates $1$ and $lambda>0$ respectively, on a graph $G$. $FPP_1$ starts from a single vertex at the origin $o$, while the initial configuration of $FPP_lambda$ consists of infinitely many seeds distributed according to a product of Bernoulli measures of parameter $mu>0$ on $V(G)setminus {o}$. $FPP_1$ starts spreading from time 0, while each seed of $FPP_lambda$ only starts spreading after it has been reached by either $FPP_1$ or $FPP_lambda$. 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 $lambda>0$ there is a $mu_0=mu_0(G,lambda)>0$ such that for all $muin(0,mu_0)$ 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 $FPP_lambda$ produces an infinite cluster almost surely for any positive $lambda,mu$, establishing fundamental differences with the behavior of such processes on $mathbb{Z}^d$.
2021
Candellero, E., DE OLIVEIRA STAUFFER, A. (2021). Coexistence of competing first passage percolation on hyperbolic graphs. ANNALES DE L'INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, in attesa di pubblicazione.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/368395
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