The branching number of intermediate growth trees

Abstract

We introduce an "intermediate branching number"(IBN) which captures the branching of intermediate growth trees, similar in spirit to the well-studied branching number of exponential growth trees. We show that the IBN is the critical parameter for recurrence vs. transience of random walk with suitably chosen (random) conductances (analogous to the results of [LP16, Chapter 3], [CHK18] for exponential and polynomial growth trees resp.). We analyze the IBN on some examples of interest. Finally we show that on some permutation wreath products of intermediate growth groups, the IBN coincides with the critical value for the firefighter problem. This gives the first tight bounds for the firefighter problem on intermediate growth groups.