Thermodynamic Analysis of Inhomogeneous Random Walks: Localization and Phase Transitions

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Abstract
We apply the thermodynamic formalism to discrete random walks in inhomogeneous environments. For simple one-dimensional examples we prove the existence of first and second order phase transitions. A typical mechanism for the latter is identified as localization-delocalization transition of relevant eigenfunctions of the generalized propagator. This can be understood by a mapping to quantum mechanical tight-binding models. Within a bivariate version of the formalism we show the occurrence of anomalous diffusion in the delocalized phase.