  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.
