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Ergodic quantum many-body systems satisfy the eigenstate thermalizationhypothesis (ETH). However, strong disorder can destroy ergodicity throughmany-body localization (MBL) -- at least in one dimensional systems -- leadingto a clear signal of the MBL transition in the probability distributions ofenergy eigenstate expectation values of local operators. For a paradigmaticmodel of MBL, namely the random-field Heisenberg spin chain, we consider thefull probability distribution of eigenstate correlation functions across theentire phase diagram. We find gaussian distributions at weak disorder, aspredicted by pure ETH. At intermediate disorder -- in the thermal phase -- wefind further evidence for anomalous thermalization in the form of heavy tailsof the distributions. In the MBL phase, we observe peculiar features of thecorrelator distributions: a strong asymmetry in $S_i^z S_{i+r}^z$ correlatorsskewed towards negative values; and a multimodal distribution for spin-flipcorrelators. A quantitative quasi-degenerate perturbation theory calculation ofthese correlators yields a surprising agreement of the full distribution withthe exact results, revealing, in particular, the origin of the multiple peaksin the spin-flip correlator distribution as arising from the resonant andoff-resonant admixture of spin configurations. The distribution of the$S_i^zS_{i+r}^z$ correlator exhibits striking differences between the MBL andAnderson insulator cases.

Statistics of correlation functions in the random Heisenberg chain