  In the probabilistic approach to quantum many-body systems, the ground-state
energy is the solution of a nonlinear scalar equation written either as a
cumulant expansion or as an expectation with respect to a probability
distribution of the potential and hopping (amplitude and phase) values recorded
during an infinitely lengthy evolution. We introduce a perturbative expansion
of this probability distribution which conserves, at any order, a
multinomial-like structure, typical of uncorrelated systems, but includes,
order by order, the statistical correlations provided by the cumulant
expansion. The proposed perturbative scheme is successfully tested in the case
of pseudo spin 1/2 hard-core boson Hubbard models also when affected by a phase
problem due to an applied magnetic field.
