Phase transition for the frog model on biregular trees

TL;DR

This paper analyzes a frog model with death on biregular trees, establishing a phase transition at a critical probability p_c, with explicit bounds and asymptotic behavior as degrees grow large.

Contribution

It provides the first rigorous analysis of phase transition behavior for the frog model with death on biregular trees, including explicit bounds for p_c.

Findings

System dies out almost surely for p < p_c

System survives with positive probability for p > p_c

p_c approaches 1/2 as degrees grow large

Abstract

We study the frog model with death on the biregular tree $\mathbb{T}_{d_1,d_2}$. Initially, there is a random number of awake and sleeping particles located on the vertices of the tree. Each awake particle moves as a discrete-time independent simple random walk on $\mathbb{T}_{d_1,d_2}$ and has a probability of death $(1-p)$ before each step. When an awake particle visits a vertex which has not been visited previously, the sleeping particles placed there are awakened. We prove that this model undergoes a phase transition: for values of $p$ below a critical probability $p_c$, the system dies out almost surely, and for $p > p_c$, the system survives with positive probability. We establish explicit bounds for $p_c$ in the case of random initial configuration. For the model starting with one particle per vertex, the critical probability satisfies $p_c(\mathbb{T}_{d_1,d_2}) = 1/2 + \Theta(1/d_1+1/d_2)$ as $d_1, d_2 \to \infty$.