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

TL;DR

This paper refines the bounds for the critical lifetime parameter in the frog model on homogeneous trees, providing tighter estimates and a comparative discussion of existing bounds using coupling methods.

Contribution

It improves the known bounds for the critical parameter p_c on both sides and offers a detailed comparison of previous results and their proofs.

Findings

Tighter bounds for p_c on both sides.

Comparison of existing bounds and proof techniques.

Use of coupling methods for analysis.

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.