Spectral diffusion in random lattices

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 $1-p$, respectively. First an exact formal solution for the decay in time of the average amplitude $〈P_0\mathrm{}\left(t\right)\mathrm{}〉$ 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 $〈P_0\mathrm{}\left(t\right)\mathrm{}〉$ at long and intermediate times are obtained close to the percolation threshold $p=p_c$, for the Bethe lattice. The solution decays as $t^{-\frac{1}{2}}$ at intermediate times and shows a long-time decay $\sim t^K\exp (-\Gamma \mathrm{}(p\left)t\right)$ towards a constant value associated with the effect of finite clusters of coupled sites. The attentuation rate is faster for $p<\mathrm{}{p}_c$ than for $p>\mathrm{}{p}_c$, 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.