2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/134875We extend the weak-coupling diagrammatic determinantal algorithm to projective schemes as well as to the inclusion of phonon degrees of freedom. The projective approach provides a very efficient algorithm to access zero temperature properties. To implement phonons, we integrate them out in favor of a retarded density-density interaction and simulate the resulting purely electronic action with the weak-coupling diagrammatic determinantal algorithm. Both extensions are tested within the dynamical mean field approximation for the Hubbard and Hubbard-Holstein models.8 pages, 4 figures, final version as appeared in PRBStrongly Correlated Electronstext