Critical random walk in random environment on a tree. Preprint. Keywords: RWRE, critical RWRE, tree, tree-indexed random walk Subject classification: Primary: 60J15. Secondary: 60G60

This paper studies the behavior of RWRE on trees in the critical case left open in previous work. For trees of exponential growth, a random perturbation of the transition probabilities can change a transient random walk into a recurrent one. This is the opposite of what occurs on trees of sub-exponential growth.