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, ${W}_{\mathrm{nm}}$, take values zero and ${W}_{0}$ with probabilities $p$ and $1\ensuremath{-}p$, respectively. First an exact formal solution for the decay in time of the average amplitude $〈{P}_{0}(t)〉$ 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 $〈{P}_{0}(t)〉$ at long and intermediate times are obtained close to the percolation threshold $p={p}_{c}$, for the Bethe lattice. The solution decays as ${t}^{\ensuremath{-}\frac{1}{2}}$ at intermediate times and shows a long-time decay $\ensuremath{\sim}{t}^{K}\mathrm{exp}(\ensuremath{-}\ensuremath{\Gamma}(p)t)$ 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.