Characterize cutoff on vertex-transitive graphs

Characterize when the simple random walk on finite vertex-transitive graphs exhibits total-variation cutoff, providing necessary and sufficient graph-structural conditions.

Background

Vertex-transitive graphs offer substantial symmetry, suggesting improved tractability. Despite major structural progress (e.g., quantitative forms of Gromov’s theorem), there is still no general result characterizing cutoff for random walk on such graphs. The notes explicitly highlight the lack of a general positive result and pose the problem of characterizing cutoff in this ensemble.