  There are several works characterizing the total-variation mixing time of a
reversible Markov chain in term of natural probabilistic concepts such as
stopping times and hitting times. In contrast, there is no known analog for the
$L_{2}$ mixing time, $\tau_{2}$ (while there are sophisticated analytic tools
to bound $ \tau_2$, in general they do not determine $\tau_2$ up to a constant
factor and they lack a probabilistic interpretation). In this work we show that
$\tau_2$ can be characterized up to a constant factor using hitting times
distributions. We also derive a new extremal characterization of the
Log-Sobolev constant, $c_{\mathrm{LS}}$, as a weighted version of the spectral
gap. This characterization yields a probabilistic interpretation of
$c_{\mathrm{LS}}$ in terms of a hitting time version of hypercontractivity. As
applications of our results, we show that (1) for every reversible Markov
chain, $\tau_2$ is robust under addition of self-loops with bounded weights,
and (2) for weighted nearest neighbor random walks on trees, $\tau_2 $ is
robust under bounded perturbations of the edge weights.
