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arXiv · 2206.13695

Critical parameter of the frog model on homogeneous trees with geometric lifetime

Sandro Gallo·Caio Pena
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Abstract

We consider the frog model with geometric lifetime (parameter $1-p$) on homogeneous trees of dimension $d$. In 2002, \cite{alves2002-2} proved that there exists a critical lifetime parameter $p_c\in(0,1)$ above which infinitely many frogs are activated with positive probability, and they gave lower and upper bounds for $p_c$. Since then, the literature on this model focussed on refinements of the upper bound. In the present paper we improve the bounds for $p_c$ \emph{on both sides}. We also provide a discussion comparing the bounds of the literature and their proofs. Our proofs are based on coupling.

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Sandro Gallo, Caio Pena. 2022-06-28. Critical parameter of the frog model on homogeneous trees with geometric lifetime. https://doi.org/10.1007/s10955-022-03049-4

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