  In the modern theory of polarization, polarization itself is given by a
geometric phase. In calculations for interacting systems the polarization and
its variance are obtained from the polarization amplitude. We interpret this
quantity as a discretized characteristic function and derive formulas for its
cumulants and moments. In the case of a non-interacting system, our scheme
leads to the gauge-invariant cumulants known from polarization theory. We study
the behavior of such cumulants for several interacting models. In a
one-dimensional system of spinless fermions with nearest neighbor interaction
the transition at which gap closure occurs can be clearly identified from the
finite size scaling exponent of the variance.
  When next nearest neighbor interactions are turned on a model with a richer
phase diagram emerges, but the finite size scaling exponent is still an
effective way to identify the localization transition.
