  Consider a sequence of continuous-time irreducible reversible Markov chains
and a sequence of initial distributions, $\mu_n$. The sequence is said to
exhibit $\mu_n$-cutoff if the convergence to stationarity in total variation
distance is abrupt, w.r.t. this sequence of initial distributions.
  In this work we give a characterization of $\mu_n$-cutoff for an arbitrary
sequence of initial distributions $\mu_n$ (in the above setup). Our
characterization is expressed in terms of hitting times of sets which are
"worst" w.r.t. $\mu_n$.
  Consider a Markov chain on $\Omega$ whose stationary distribution in $\pi$.
Let $t_{\mathrm{H}}(\alpha) :=\max_{x \in \Omega,A \subset \Omega :\,\pi(A) \ge
\alpha}\mathbb{E}_{x}[T_{A}]$ be the expected hitting time of the worst set of
size at least $\alpha$. It was recently proved by Peres and Sousi and
independently by Oliveira that $t_{\mathrm{H}}(1/4) $ captures the order of the
mixing time. In this work we further refine this connection and show that
$\mu_n$-cutoff can be characterized in terms of concentration of hitting times
(starting from $\mu_n$) of sets which are worst in expectation w.r.t. $\mu_n$.
Conversely, we construct a counter-example which demonstrates that in general
cutoff (as opposed to cutoff w.r.t. a certain sequence of initial
distributions) cannot be characterized in this manner.
  Finally, we also prove that there exists an absolute constant $C$ such that
for every Markov chain $\epsilon(
t_{\mathrm{H}}(\epsilon)-t_{\mathrm{H}}(1-\epsilon)) \le Ct_{\mathrm{rel}}
|\log \epsilon|$, for all $0< \epsilon < 1/2$, where $t_{\mathrm{rel}} $ is the
inverse of the spectral gap of the chain.
