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

Geometric branching operators on the Bruhat-Tits building associated to the symplectic group