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.

References

In particular, proving cutoff on vertex-transitive expanders is one of the most famous open problems in the field (see [Open Question 34] or [Question 5]).

— Modern aspects of Markov chains: entropy, curvature and the cutoff phenomenon  (2508.21055 - Salez, 28 Aug 2025) in Section 2.1, Expander graphs

There is a intriguing conjecture that every vertex-transitive expander exhibits cutoff, which remains open Conjecture~1.7.

— Repeated averaging on expanders I: random $d$-regular graph  (2609.25729 - Yao et al., 22 Sep 2026) in Section 1, Introduction, paragraph discussing sparse expander random walks