The recurrence–transience conjecture for the frog model on rooted trees

Let $\mathbb{T}_d$ be the infinite rooted $d$-ary tree, and call a realization of the frog model recurrent if the root is visited infinitely many times and transient if the root is visited finitely often.

Recurrence–transience conjecture. The frog model on $\mathbb{T}_d$ is recurrent almost surely for $d=3$ and transient almost surely for $d=4$.

The recurrence and transience behavior is known for all other degrees covered by the main theorem: the model is almost surely recurrent on $\mathbb{T}_2$ and almost surely transient on $\mathbb{T}_d$ for $d\geq 5$. The cases $d=3$ and $d=4$ were left unresolved and supported by simulation evidence.