  In this thesis we study geometric branching operators on simplices in the
(affine) Bruhat-Tits building associated to the symplectic group over a local
field. We use these operators to study the simple random walk on the building's
vertices. This work lays the foundations for studying the total-variation
cutoff phenomenon for simple random walk on Ramanujan complexes of type
$\tilde{C}_n$ ($n\geq 2$).
