  We study the ergodic side of the many-body localization transition in its
standard model, the disordered Heisenberg quantum spin chain. We show that the
Thouless energy, extracted from long-range spectral statistics and the
power-spectrum of the full momentum distribution fluctuations, is not large
enough to guarantee thermalization. We find that both estimates coincide and
behave non-monotonically, exhibiting a strong peak at an intermediate value of
the disorder. Furthermore, we show that non-thermalizing initial conditions
occur well within the ergodic phase with larger probability than expected.
Finally, we propose a mechanism, driven by the Thouless energy and the presence
of anomalous events, for the transition to the localized phase.
