In a weakly disordered sample, light waves (or electrons) propagate, on average, by diffusion. However, with some small probability, random high-quality cavities can be formed within the sample. Such cavities are due to rare events, i.e., to some rare disorder configurations which can support “almost localized” eigenstates and thus can trap the wave for a long time in a small region of space of sub-mean-free-path size. The almost localized states are nonuniversal in the sense that their character and likelihood are determined not only by the average strength of the disorder (the dimensionless conductance) but also by microscopic details of the system. In particular, they are extremely sensitive to the correlation radius of the disordered potential. Moreover, on a lattice, a new type of almost localized state becomes possible that has no analog in the continuum. The likelihood of these lattice-specific states decreases with the increase of in sharp contrast to the situation in the continuum. We review the earlier work on the almost localized states in the continuum and develop a theory of those states on a lattice. We emphasize that extreme care must be taken in trying to simulate on a lattice the rare events in a continuous, random medium.
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