Restart perturbations for reversible Markov chains: trichotomy and pre-cutoff equivalence

Given a reversible Markov chain $P_n$ on $n$ states, and another chain $\tilde{P}_n$ obtained by perturbing each row of $P_n$ by at most $\alpha_n$ in total variation, we study the total variation distance between the two stationary distributions, $\| \pi_n - \tilde{\pi}_n \|$. We show that for chains with cutoff, $\| \pi_n - \tilde{\pi}_n \|$ converges to $0$, $e^{-c}$, and $1$, respectively, if the product of $\alpha_n$ and the mixing time of $P_n$ converges to $0$, $c$, and $\infty$, respectively. This echoes recent results for specific random walks that exhibit cutoff, suggesting that cutoff is the key property underlying such results. Moreover, we show $\| \pi_n - \tilde{\pi}_n \|$ is maximized by restart perturbations, for which $\tilde{P}_n$ "restarts" $P_n$ at a random state with probability $\alpha_n$ at each step. Finally, we show that pre-cutoff is (almost) equivalent to a notion of "sensitivity to restart perturbations," suggesting that chains with sharper convergence to stationarity are inherently less robust.