Cutoff on Ramanujan complexes and classical groups

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 sparse graphs in general. In paper we establish Ramanujan complexes of type $\widetilde{A}_{d}$ $(d\geq1)$. As a result, obtain explicit generators the finite classical groups $\mathrm{PGL}_{n}(\mathbb{F}_{q})$ which associated Cayley exhibit cutoff.