Spectral diffusion in random lattices

Abstract

The classical diffusion of localized excitations is studied on random linear chain and Bethe lattices (connectivity K) in which the nearest-neighbor transfer rates, W_nm, take values zero and W₀ with probabilities p and 1-p, respectively. First an exact formal solution for the decay in time of the average amplitude <P₀(t)> of an initial excitation at a lattice site is discussed, using the analogy between the diffusion problem and the response of a random impedance network to a localized current pulse. Detailed results for <P₀(t)> at long and intermediate times are obtained close to the percolation threshold p=p_c, for the Bethe lattice. The solution decays as t^-½ at intermediate times and shows a long-time decay ~t^Kexp(-Γ(p)t) towards a constant value associated with the effect of finite clusters of coupled sites. The attentuation rate is faster for p<p_c than for p>p_c, as expected. The one-dimensional case requires a special treatment which is shown to give results identical to those of a different earlier analysis. The generality of our method suggests its application to various other problems.