  The cutoff phenomenon was recently confirmed for random walks on Ramanujan
graphs by the first author and Peres. In this work, we obtain analogs in higher
dimensions, for random walk operators on any Ramanujan complex associated with
a simple group $G$ over a local field $F$. We show that if $T$ is any
$k$-regular $G$-equivariant operator on the Bruhat-Tits building with a simple
combinatorial property (collision-free), the associated random walk on the
$n$-vertex Ramanujan complex has cutoff at time $\log_k n$. The high
dimensional case, unlike that of graphs, requires tools from non-commutative
harmonic analysis and the infinite-dimensional representation theory of $G$.
Via these, we show that operators $T$ as above on Ramanujan complexes give rise
to Ramanujan digraphs with a special property ($r$-normal), implying cutoff.
Applications include geodesic flow operators, geometric implications, and a
confirmation of the Riemann Hypothesis for the associated zeta functions over
every group $G$, previously known for groups of type $\widetilde A_n$ and
$\widetilde C_2$.
