Cutoff on vertex-transitive expander graphs

Prove that the simple random walk on every finite vertex-transitive expander graph exhibits total-variation cutoff.

Background

Expanders are bounded-degree graphs with spectral gaps bounded away from zero; they mix in time Θ(log |V|) and satisfy the product condition. Cutoff is established for some highly structured expanders (e.g., Ramanujan) but remains unknown in general. The notes highlight this target as one of the most famous open problems in the area.