Stable limit laws for randomly biased walks on supercritical trees

Abstract

We consider a random walk on a supercritical Galton–Watson tree with leaves, where the transition probabilities of the walk are determined by biases that are randomly assigned to the edges of the tree. The biases are chosen independently on distinct edges, each one according to a given law that satisfies a logarithmic nonlattice condition. We determine the condition under which the walk is sub-ballistic, and, in the sub-ballistic regime, we find a formula for the exponent $\gamma \in(0,1)$ such that the distance $\vert X(n)\vert$ moved by the walk in time $n$ is of the order of $n^{\gamma }$. We prove a stable limiting law for walker distance at late time, proving that the rescaled walk $n^{-\gamma }\vert X(n)\vert$ converges in distribution to an explicitly identified function of the stable law of index $\gamma $.

This paper is a counterpart to Ben Arous et al. [Ann. Probab. 40 (2012) 280–338], in which it is proved that, in the model where the biases on edges are taken to be a given constant, there is a logarithmic periodicity effect that prevents the existence of a stable limit law for scaled walker displacement. It is randomization of edge-biases that is responsible for the emergence of the stable limit in the present article, while also introducing further correlations into the model in comparison with the constant bias case. The derivation requires the development of a detailed understanding of trap geometry and the interplay between traps and backbone. The paper may be considered as a sequel to Ben Arous and Hammond [Comm. Pure Appl. Math. 65 (2012) 1481–1527], since it makes use of a result on the regular tail of the total conductance of a randomly biased subcritical Galton–Watson tree.