%0 Journal Article %T Almost sure convergence for stochastically biased random walks on trees %A Gabriel Faraud %A Yueyun Hu %A Zhan Shi %J Mathematics %D 2010 %I arXiv %R 10.1007/s00440-011-0379-y %X We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, 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. %U http://arxiv.org/abs/1003.5505v5