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

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.