Optimal reward on a sparse tree with random edge weights

It is well known that the maximal displacement of a random walk indexed by an m-ary tree with bounded independent and identically distributed edge weights can reliably yield much larger asymptotics than a classical random walk whose summands are drawn from the same distribution. We show that, if the edge weights are mean-zero, then nonclassical asymptotics arise even when the tree grows much more slowly than exponentially. Our conditions are stated in terms of a Minkowski-type logarithmic dimension of the boundary of the tree.