  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 non-lattice 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 (which is positive but less than one) such that
the distance | X_n | 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} | X_n | converges in distribution to an explicitly
identified function of the stable law of index gamma. This paper is a
counterpart to [4], 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 [2], since it makes use of a result on the regular
tail of the total conductance of a randomly biased subcritical Galton-Watson
tree.
