  On the grounds of a Feynman-Kac--type formula for Hamiltonian lattice systems
we derive analytical expressions for the matrix elements of the evolution
operator. These expressions are valid at long times when a central limit
theorem applies. As a remarkable result we find that the ground-state energy as
well as all the correlation functions in the ground state are determined
semi-analytically by solving a simple scalar equation. Furthermore, explicit
solutions of this equation are obtained in the noninteracting case.
