 Shift-refresh chain on a hypercube

A homogeneous shift-refresh chain on a dimension-$m$ hypercube has state $x=(x_1,\dots ,x_m)\in \{0,1{\}}^m$ and transitionwhere each fresh $U$ is an independent fair Bernoulli random variable. It rotates the old coordinates and refreshes the final one. The uniform distribution on bit vectors is stationary, because shifting iid fair bits and appending another fair bit preserves their joint law.

After exactly $m$ steps, every coordinate consists of a distinct fresh fair bit, independently of the starting state. Thus the deterministic Stopping time $\tau =m$ is a strong stationary time. Starting at a fixed vector, at time $t<m$ there are $m-t$ deterministic retained coordinates and $t$ independent fair coordinates. The law is uniform on a subset of $2^t$ states, giving the exact total variation distanceConsequently the family has cutoff at $m$ with a bounded window: at $(1-\epsilon )m$ the distance tends to one for every fixed $\epsilon >0$, whereas at $m$ it is zero.

A pointer description must specify which variables belong to the state. If a pointer itself is included in an augmented chain, refreshing all bits makes the bits uniform conditional on the pointer history, but does not by itself make the pointer stationary or independent of the Stopping time. Coverage alone is therefore insufficient to establish a strong stationary time for that augmented chain.