Deterministic structural trigger for cutoff on random graphs (derandomization)

Identify a deterministic structural property that is almost surely satisfied by standard sparse random graph models (such as random regular graphs or related sparse ensembles) and that, when verified on a specific graph instance, deterministically implies total-variation cutoff for the simple random walk on that graph.

Background

Recent advances exploit the tree-like geometry of sparse random graphs to prove cutoff for simple random walk in various models. However, these proofs are tailored to randomness and do not provide a general deterministic structure that guarantees cutoff. The notes call for a structural condition that can be checked deterministically and that these random graphs satisfy with high probability.