  In systems with a large degeneracy of states such as black holes, one expects
that the average value of probe correlation functions will be well approximated
by the thermal ensemble. To understand how correlation functions in individual
microstates differ from the canonical ensemble average and from each other, we
study the variances in correlators. Using general statistical considerations,
we show that the variance between microstates will be exponentially suppressed
in the entropy. However, by exploiting the analytic properties of correlation
functions we argue that these variances are amplified in imaginary time,
thereby distinguishing pure states from the thermal density matrix. We
demonstrate our general results in specific examples and argue that our results
apply to the microstates of black holes.
