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Spreading, Nonergodicity, and Selftrapping: a puzzle of interacting disordered lattice waves

Localization of waves by disorder is a fundamental physical problem encompassing a diverse spectrum of theoretical, experimental and numerical studies in the context of metal-insulator transitions, the quantum Hall effect, light propagation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays, to name just a few examples. Large intensity light can induce nonlinear response, ultracold atomic gases can be tuned into an interacting regime, which leads again to nonlinear wave equations on a mean field level. The interplay between disorder and nonlinearity, their localizing and delocalizing effects is currently an intriguing and challenging issue in the field of lattice waves. In particular it leads to the prediction and observation of two different regimes of destruction of Anderson localization - asymptotic weak chaos, and intermediate strong chaos, separated by a crossover condition on densities. On the other side approximate full quantum interacting many body treatments were recently used to predict and obtain a novel many body localization transition, and two distinct phases - a localization phase, and a delocalization phase, both again separated by some typical density scale. We will discuss selftrapping, nonergodicity and nonGibbsean phases which are typical for such discrete models with particle number conservation and their relation to the above crossover and transition physics. We will also discuss potential connections to quantum many body theories.