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 Levy distribution that grows up algebraically with an universal exponent (2+d)/theta for d less than or equal tod(c)=2/(theta -1). For large distances the probability distribution decays exponentially on a characteristic length scale xi similar or equal to (D/lambda)(1/[2+d(1-theta)]). Above d(c) only the diffusion regime remains. The relation to electron localization is discussed.