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We study the propagation of lattice vibrations in models of disordered, classical anharmonic crystals. Using classical perturbation theory with an optimally chosen remainder term (i.e. a Nekhoroshev-type scheme), we are able to show that vibrations corresponding to localized initial conditions do essentially not propagate through the crystal up to times larger than any inverse power of the strength of the anharmonic couplings

A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom