  The total-variation cutoff phenomenon has been conjectured to hold for simple
random walk on all transitive expanders. However, very little is actually known
regarding this conjecture, and cutoff on sparse graphs in general. In this
paper we establish total-variation cutoff for simple random walk on Ramanujan
complexes of type $\widetilde{A}_{d}$ $(d\geq1)$. As a result, we obtain
explicit generators for the finite classical groups
$\mathrm{PGL}_{n}(\mathbb{F}_{q})$ for which the associated Cayley graphs
exhibit total-variation cutoff.
