Random-walk theory for localization

A continuous-time random-walk theory has been developed for Anderson localization. On a continuous time scale random walks are performed along extended (i.e., propagating) and localized (i.e., trap) states. Complete information of disorder is contained in a distribution function called “hopping time distribution function” ψ_nm(t), which gives the probability per unit time for transition from state m to state n in time t. The “stay-put” probability ??(t = ∞), which is the probability to rediscover an excitation at a site “0” at time t = ∞ if it was there at t = 0, is obtained in terms of ψ_nm(t). Appropriate forms for ψ_nm(t) are constructed which are in conformity with the photoconductivity experiments on dispersive transport, and ??(∞) are calculated. The results indicate that the entire spectrum consists of three regimes, namely, those of (i) “diffusion,” (ii) “weak diffusion,” and (iii) “no diffusion,” which, respectively, designate the extension, the power-law localization, and the exponential localization of states. The results also shed light on the question of “continuous or discontinuous (?)” transition across the mobility edge.