  We suggest and implement a new Monte Carlo strategy for correlated models
involving fermions strongly coupled to classical degrees of freedom, with
accurate handling of quenched disorder as well. Current methods iteratively
diagonalise the full Hamiltonian for a system of N sites with computation time
tau_N proportional to N^4. This limits achievable sizes to N \sim 100. In our
method the energy cost of a Monte Carlo update is computed from the Hamiltonian
of a cluster, of size N_c, constructed around the reference site, and embedded
in the larger system. As MC steps sweep over the system, the cluster
Hamiltonian also moves, being reconstructed at each site where an update is
attempted. In this method tau_{N,N_c} is proportional to NN_c^3. Our results
are obviously exact when N_c=N, and converge quickly to this asymptote with
increasing N_c. The accuracy improves in systems where the effective disorder
seen by the fermions is large. We provide results of preliminary calculations
on the Holstein model and the Double Exchange model. The `locality' of the
energy cost, as evidenced by our results, suggests that several important but
inaccessible problems can now be handled with control.
