Phase Transition for the Frog Model on Biregular Trees

#### E. Lebensztayn, J. Utria

2020, v.26, Issue 3, 447-466

ABSTRACT

We study the \emph{frog model with death\/} on the biregular tree $\mathbb{T}_{d_1,d_2}$.
Initially, there is a random number of active and inactive particles located on the vertices of the tree.
Each active 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 active particle visits a vertex which has not been visited previously, the inactive particles placed there are activated.
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 $d_1, d_2 \to \infty$.

Keywords: frog model, biregular tree, phase transition