RANDOM WALK APPROACH TO THE ANALYTICAL SOLUTION OF DISORDERD SYSTEMS WITH MULTIPLICATIVE NOISE - THE ANDERSON LOCALIZATION PROBLEM
V. Kuzovkov,
W. von Niessen (Braunschweig Technical University, Germany).


Understanding fundamental properties of low-dimension disorded systems continue to attract great attention. In collaboration with Technische Universitat Braunschweig, Germany, we developed a new analytical random walk approach for calculating the phase-diagram of spatially extended systems with multiplicative noise. We study the Anderson localization problem as an example. The transition from delocalized to localized states was treated as a generalized diffusion with a noise-induced first-order phase transition. The generalized diffusion manifests itself in the divergence of averages of wavefunctions (correlators) controlled by the Lyapunov exponent, which is the inverse of the localization length. The appearance of the generalized diffusion arises due to the instability of a fundamental mode corresponding to correlators. The generalized diffusion could be described in terms of a signal theory, which operates with the concepts of input and output signals and the filter function. Delocalized states correspond to the bounded output signals whereas localized states to unbounded output signals, respectively. The transition from bounded to unbounded signals is defined uniquely by the filter function.


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