  Closed, interacting, quantum systems have the potential to transition to a
many-body localized (MBL) phase under the presence of sufficiently strong
disorder, hence breaking ergodicity and failing to thermalize. In this work we
study the distribution of correlations throughout the ergodic-MBL phase
diagram. We find the typical correlations in the MBL phase decay as a stretched
exponential with range $r$ eventually crossing over to an exponential decay
deep in the MBL phase. At the transition, the stretched exponential goes as
$e^{-A\sqrt{r}}$, a decay that is reminiscent of the random singlet phase.
While the standard deviation of the $\log(QMI)$ has a range dependence, the
$\log(QMI)$ converges to a range-invariant distribution on all other moments
(i.e., the skewness and higher) at the transition. The universal nature of
these distributions provides distinct phenomenology of the transition different
from both the ergodic and MBL phenomenologies. In addition to the typical
correlations, we study the extreme correlations in the system, finding that the
probability of strong long-range correlations is maximal at the transition,
suggesting the proliferation of resonances there. Finally, we analyze the
probability that a single bit of information is shared across two halves of a
system, finding that this probability is non-zero deep in the MBL phase but
vanishes at moderate disorder well above the transition.
