Phase transition for the frog model on biregular trees

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 $p_c(\mathbb{T}_{d_1,d_2}) = 1/2 + Θ(1/d_1+1/d_2)$ as $d_1, d_2 \to \infty$.

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