Anderson localization on the Bethe lattice
Anderson localization - the suppression of wave propagation by interference in a sufficiently disordered medium - is one of the most remarkable phenomena in condensed matter physics. These lecture notes provide a pedagogical introduction to Anderson localization on the Bethe lattice, an infinite tree with fixed connectivity and no loops, and one of the few settings in which the transition can be studied exactly. The problem is important for two main reasons. First, the Anderson model on the Bethe lattice provides a mean-field, infinite-dimensional limit in which the transition point and critical behavior can be determined exactly. Second, it provides a simplified description of the high-dimensional configuration spaces of interacting quantum systems, and thus a useful toy model for many-body localization. We focus on the Green's function formalism and the cavity method. After recalling standard diagnostics of localization based on transport, spectral statistics, and eigenfunction moments, we show that they are encoded in the statistics of the Green's functions, and that the full distribution of the local density of states is the natural order parameter of the transition. We derive the exact self-consistent cavity equations on the Bethe lattice, use them to characterize the two phases and critical behavior - including an analytic solution at large connectivity - and describe population-dynamics algorithms that solve them with high precision. We explain how an exponentially large correlation volume and the localized nature of the critical point account for the strong finite-size effects on random regular graphs, which led to a long debate over a possible intermediate non-ergodic phase. Finally, we discuss the mapping onto directed polymers in random media, interpreting localization as a freezing transition dominated by a few rare resonant paths.