Stable limit laws for randomly biased walks on supercritical trees

TL;DR

This paper studies a random walk on a supercritical Galton-Watson tree with randomly assigned edge biases, establishing conditions for sub-ballistic behavior and proving a stable limit law for the walk's scaled displacement.

Contribution

It introduces a model with random edge biases, derives the sub-ballistic regime, and proves a stable limit law for the walk's displacement, highlighting the role of randomness in bias.

Findings

Identifies conditions for sub-ballistic behavior.

Derives a formula for the exponent gamma.

Proves convergence to a stable law for scaled displacement.

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 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.