  Dynamics of the imbalance in occupations on even and odd sites of a lattice
serves as one of the key characteristics for identification of the many-body
localization transition. In this work, we investigate the long-time behaviour
of the imbalance in disordered one- and two-dimensional many-body systems in
the regime of diffusive or subdiffusive transport. We show that memory effects
originating from a coupling between slow and fast modes lead to a power-law
decay of the imbalance, with the exponent determined by the diffusive (or
subdiffusive) transport law and the spatial dimensionality. Analytical results
are supported by numerical simulations performed on a two-dimensional system in
the regime of weak localization.
