  A multi-type branching process is introduced to mimic the evolution of the
avalanche activity and determine the critical density of the Abelian Manna
model. This branching process incorporates partially the spatio-temporal
correlations of the activity, which are essential for the dynamics, in
particular in low dimensions. An analytical expression for the critical density
in arbitrary dimensions is derived, which significantly improves the results
over mean-field theories, as confirmed by comparison to the literature on
numerical estimates from simulations. The method can easily be extended to
lattices and dynamics other than those studied in the present work.
