Randomly biased walks on subcritical trees

Abstract

As a model of trapping by biased motion in random structure, we study the time taken for a biased random walk to return to the root of a subcritical Galton-Watson tree. We do so for trees in which these biases are randomly chosen, independently for distinct edges, according to a law that satisfies a logarithmic non-lattice condition. The mean return time of the walk is in essence given by the total conductance of the tree. We determine the asymptotic decay of this total conductance, finding it to have a pure power-law decay. In the case of the conductance associated to a single vertex at maximal depth in the tree, this asymptotic decay may be analysed by the classical defective renewal theorem, due to the non-lattice edge-bias assumption. However, the derivation of the decay for total conductance requires computing an additional constant multiple outside the power-law that allows for the contribution of all vertices close to the base of the tree. This computation entails a detailed study of a convenient decomposition of the tree, under conditioning on the tree having high total conductance. As such, our principal conclusion may be viewed as a development of renewal theory in the context of random environments.For randomly biased random walk on a supercritical Galton-Watson tree with positive extinction probability, our main results may be regarded as a description of the slowdown mechanism caused by the presence of subcritical trees adjacent to the backbone that may act as traps that detain the walker. Indeed, this conclusion is exploited in [18] to obtain a stable limiting law for walker displacement in such a tree.

Phase Transition for the Speed of the Biased Random Walk on the Supercritical Percolation Cluster

“…In the context of biased random walk, the infinite tree may be considered to be a backbone, on which the walk advances at linear speed, while the subcritical trees are the traps that may delay the walk. The papers [13], [16] and [35] undertake a detailed examination of trapping for biased random walks on such trees. While involved, these works depend in an essential way on the unambiguous backbone-trap decomposition available for such environments.…”

Lyons‐Pemantle‐Peres Monotonicity Problem for High Biases

“…On Galton-Watson trees with leaves (see [2,3,10,13]) or on supercritical percolation clusters (see [5,8,16]), the speed is certainly not increasing since it eventually vanishes. In these models, the slowdown of the walk can be explained by the presence of dead ends in the environment, which act as powerful traps.…”

“…Let us explain what the difficulty of this problem is since, indeed, questions about the speed of random walks in random environments can be subtle (see [17,18,20] for general reviews of the subject). On Galton-Watson trees with leaves (see [2,3,10,13]) or on supercritical percolation clusters (see [5,8,16]), the speed is certainly not increasing since it eventually vanishes. In these models, the slowdown of the walk can be explained by the presence of dead ends in the environment, which act as powerful traps.…”

Biased random walk in positive random conductances on $\mathbb{Z}^{d}$

“…One very characteristic behavior associated to trapping is the existence of a zero asymptotic speed for RWRE with directional transience. In the last few years, several articles have analyzed such models from a trapping perspective, such as [13] and [14] on Z and [2,3] and [17] on trees. The results on the d-dimensional lattice (with d ≥ 2) are much more rare, since So far, mathematically, only two models of biased random walks in Z d have been studied from a trapping perspective: one is on a supercritical percolation cluster and the other is in environments assumed to be uniformly elliptic.…”