  We review a recent approach for the simulation of many-body interacting
systems based on an efficient generalization of the Lanczos method for Quantum
Monte Carlo simulations. This technique allows to perform systematic
corrections to a given variational wavefunction, that allow to estimate exact
energies and correlation functions, whenever the starting variational
wavefunction is a qualitatively correct description of the ground state. The
stability of the variational wavefunction against possible phases, not
described at the variational level can be tested by using the ''effective
Hamiltonian'' approach. In fact Monte Carlo methods, such as the ''fixed node
approximation'' and the ''generalized Lanczos technique'' (Phys. Rev. B
64,024512, 2001) allow to obtain exact ground state properties of an effective
Hamiltonian, chosen to be as close as possible to the exact Hamiltonian, thus
yielding the most reasonable estimates of correlation functions. We also
describe a simplified one-parameter scheme that improve substantially the
efficiency of the generalized Lanczos method. This is tested on the t-J model,
with a special effort to obtain accurate pairing correlations, and provide a
possible non-phonon mechanism for High temperature superconductivity.
