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 $\tilde{C}_n$ ($n\geq 2$).