Dynamics of a harmonic oscillator on the Bethe lattice

Abstract

The velocity autocorrelation functions for a classical coupled harmonic oscillator on the Bethe lattice are exactly evaluated with use of the continued fraction formalism. A long-time tail of ${t}^{\ensuremath{-}3/2}$ leading to a vanishing diffusion coefficient results from the localized excitations with a gap occurring due to the nonexistence of a well-defined wave vector. The strongly colored fluctuating forces in the generalized Langevin equation are specified by the memory functions with a tail of ${t}^{\ensuremath{-}3/2}.$