Localization transition and ensemble inequivalence in two paradigmatic cases: the Discrete Non-Linear Schrödinger Equation and the ideal Bose-Einstein condensation

We review here the underlying mechanism of the localization transition in two paradigmatic cases where, due to the simplicity of the systems, a large-deviation approach can be carried on in full detail, revealing in both cases the first-order features of this transition. The key ingredient is the calculation of subleading contributions to the free-energy/free-entropy. Among the most recent results we will illustrate the non-trivial behaviour of entanglement entropy for the localization transition in the Discrete Nonlinear Schrodinger Equation and the precise characterization of ideal Bose-Einstein condensation as a a large-deviation phenomenon in the canonical ensemble. 

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