Almost sure convergence for stochastically biased random walks on trees

We are interested in the biased random walk on a supercritical Galton?CWatson tree in the sense of Lyons (Ann. Probab. 18:931?C958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields 106:249?C264, 1996), and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system??s non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)³, converges almost surely to a known positive constant.