Eichler orders, quotient graphs and random walks

We study the extent to which the quotient of the Bruhat-Tits tree at one place $Q$, associated to a genus of orders of maximal rank, can be computed from the analogous quotient at a different place $P$. We show that this computation can be carried out, except for a small set of vertices depending on $P$, but not on $Q$. We give some geometrical conditions on the quotient at $P$ that ensure that this exceptional set is empty. This generalizes the formulas from a previous work that allow the computation of the quotient graph at all places, for the genus of maximal orders over the projective line. The methods presented here yield similar results for other genera or other curves.