The oldest model of random trees is that of Galton and Watson. In joint work with Robin Pemantle and Yuval Peres, described informally here, we consider simple random walk on the family tree T of a supercritical Galton-Watson branching process. First we calculate the speed (i.e., rate of escape) of the walk; the answer is simple, yet surprising. Then we show that the resulting harmonic measure has a.s.\ strictly smaller dimension than that of the whole boundary of T. Concretely, this means that after T is picked but before the random walk takes place, a subtree of T of a smaller exponential growth rate can be specified such that with overwhelming probability, the random walk particle will never exit this subtree (!).



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