Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 215 / 4 / a / Solution

For $0<\alpha ,\epsilon <1$, defineA hit${}_{\alpha }$ cutoff means that for every fixed $0<\epsilon <1/2$,For finite reversible chains, mixing-time cutoff implies hit${}_{1/2}$ cutoff; this is the hitting-time characterization of cutoff.