Feedback-induced localization in random walks

The random walk of a single particle with diffusion constant D is considered under the influence of a self-organized feedback coupling of strength $\lambda$ to the environment of the particle. Assuming that the memory kernel, responsible for the feedback, has a power-law behavior with an exponent $\theta$ the particle will be localized near the origin. Around that region the stationary probability density leads to a Lévy distribution that grows up algebraically with an universal exponent $\mathrm{}(2+d)/\theta$ for $d<~d_c=2/\left(\theta -1\right).$ For large distances the probability distribution decays exponentially on a characteristic length scale $\xi \simeq (D/\lambda {)}^{1/[2+d(1-\theta \left)\right]}.$ Above $d_c$ only the diffusion regime remains. The relation to electron localization is discussed.