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The motivation for this paper is the study of the phase transition forrecurrence/transience of a class of self-interacting random walks on trees,which includes the once-reinforced random walk. For this purpose, we define aquantity, that we call the branching-ruin number of a tree, which provides (inthe spirit of Furstenberg, 1970, and Lyons, 1990) a natural way to measuretrees with polynomial growth. We prove that the branching-ruin number of a treeis equal to the critical parameter for the recurrence/transience of theonce-reinforced random walk. We define a sharp and effective (i.e. computable)criterion characterizing the recurrence/transience of a larger class ofself-interacting walks on trees, providing the complete picture for their phasetransition.

The branching-ruin number and the critical parameter of once-reinforced random walk on trees