Almost sure convergence for stochastically biased random walks on trees

Abstract

We are interested in the biased random walk on a supercritical Galton--Watson tree in the sense of Lyons, Pemantle and Peres, and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system's non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)^3, converges almost surely to a known positive constant.

References in corpus (4)

• Transient Random Walks in Random Environment on a Galton-Watson Tree
• Large deviations for random walks in random environment on a Galton-Watson tree
• Biased random walks on a Galton-Watson tree with leaves
• A central limit theorem for random walk in random environment on marked Galton-Watson trees

Cited by in corpus (14)

• Maximal displacement of a branching random walk in time-inhomogeneous environment
• Branching random walk with selection at critical rate
• The maximal potential energy of biased random walks on trees
• The slow regime of randomly biased walks on trees
• Einstein relation for biased random walk on Galton--Watson trees
• Local times of subdiffusive biased walks on trees
• The most visited sites of biased random walks on trees
• Generalized range of slow random walks on trees
• Stable limit laws for randomly biased walks on supercritical trees
• Randomly biased walks on subcritical trees
• Fine asymptotics for the consistent maximal displacement of branching Brownian motion
• Necessary and sufficient conditions for the convergence of the consistent maximal displacement of the branching random walk
• Coalescence in small generations for the diffusive randomly biased walk on Galton-Watson trees
• Consistent Minimal Displacement of Branching Random Walks