Random walks in I.I.D. random environment on Cayley trees

We consider the random walk in an i.i.d. random environment on the infinite d-regular tree for d≥3. We consider the tree as a Cayley graph of the free product of finitely many copies of Z and Z2 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.