  Characterizing the nature of many-body localization transitions (MBLTs) and
their potential critical behaviors has remained a challenging problem. In this
work, we study the dynamics of the displacement, quantifying the spread of the
radial probability distribution in the Fock space, for systems with MBLTs, and
perform a finite-size scaling analysis. We find that the scaling exponents
satisfy theoretical bounds, and can identify universality classes. We show that
reliable extrapolations to the thermodynamic limit for the MBLT induced by
quasiperiodic fields is possible even for computationally accessible system
sizes. Our work highlights that the displacement is a valuable tool for
studying MBLTs, as relevant to ongoing experimental efforts.
