  We introduce new diagnostics of the transition between the ergodic and
many-body localization phases, which are based on complexity measures defined
via the probability distribution function of the Lanczos coefficients of the
tri-diagonalized Hamiltonian. We use these complexity measures to analyze the
power-law random banded matrix model as a function of the correlation strength
and show that the moments and the entropy of the distribution diagnose the
ergodic to many-body transition, as well as the distinctive feature of the
phases concerning the memory of the initial conditions.
