Random Walks in I.I.D. Random Environment on Cayley Trees

TL;DR

This paper studies the behavior of random walks in i.i.d. random environments on infinite Cayley trees, proving transience under mild conditions, and connects the environment's invariance to the group's structure.

Contribution

It demonstrates that random walks in i.i.d. environments on Cayley trees are always transient, extending understanding of walk behavior in complex group-invariant environments.

Findings

Random walks are always transient under mild assumptions.

Environment invariance under group action influences walk behavior.

Connects group structure with probabilistic properties of walks.

Abstract

We consider the random walk in an \emph{i.i.d.} random environment on the infinite $d$-regular tree for $d \geq 3$. We consider the tree as a Cayley graph of free product of finitely many copies of $\Zbold$ and $\Zbold_2$ and define the i.i.d. environment as invariant under the action of this group. Under a mild non-degeneracy assumption we show that the walk is always transient.