For $0<\alpha,\varepsilon<1$, define
$$
\operatorname{hit}_\alpha(\varepsilon)
=\inf\left\{t:\max_x\max_{A:\pi(A)\geq\alpha}
\mathbb P_x(T_A>t)\leq\varepsilon\right\}.
$$
A hit$_\alpha$ cutoff means that for every fixed $0<\varepsilon<1/2$,
$$
\frac{\operatorname{hit}_\alpha(\varepsilon)}
{\operatorname{hit}_\alpha(1-\varepsilon)}\to1.
$$
For finite reversible chains, mixing-time cutoff implies hit$_{1/2}$ cutoff; this is the hitting-time characterization of cutoff.

Solution

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

 Solution

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