Recurrence for the simple frog model on T3 and T4

Determine whether the standard frog model without death (q = 1), in which one active frog starts at the root of the infinite rooted d-ary tree T_d and all other vertices contain one sleeping frog that performs simple random walk upon activation, is recurrent with positive probability for d = 3 and d = 4 (i.e., whether the root is visited by infinitely many active frogs with positive probability).

Background

The paper studies recurrence and transience of the frog model on trees and variants with death and drift. In this context, recurrence means the root is visited by infinitely many active frogs. The classical case with no death (q = 1) and no drift is particularly challenging.

While recurrence with positive probability for the binary tree (T2) was eventually established after a long effort, the analogous question for the ternary and quaternary trees (T3 and T4) remains unresolved. This highlights the difficulty of proving recurrence even in seemingly simple tree settings and motivates the development of new techniques such as the recursive frog model considered in this work.