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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 indoi:10.1088/1367-2630/6/1/107 fatcat:nlzwdihv6jdp3cwghsppdybeh4