Low-frequency vibrational spectrum and low-temperature specific heat of Bethe lattices.

Abstract

By mapping the harmonic-oscillations equations onto the random-walk equations, we calculate analytically the low-frequency vibrational spectrum of a Bethe lattice and show that it has a gap between zero frequency and a critical frequency. Using these results, we then obtain a low-temperature asymptotic expression for the vibrational specific heat and find that it goes exponentially to zero as T\ensuremath{\rightarrow}0.