Loop expansion around the Bethe solution for disordered models.

Recently we have introduced a new loop expansion around the Bethe solution that could be applied to any model well defined on a Bethe lattice. This new expansion is an alternative to the standard field-theoretical renormalization group approach, which can be viewed as a loop expansion around fully connected mean-field models. We will illustrate the application of this new method in understanding the phase transitions undergone by different disordered systems and emphasize its possible application to models where the standard expansion does not work, for example, Anderson localization or (bootstrap) percolation. We will also illustrate the progress achieved by the use of this new expansion in the characterization of avalanches in the Random Field Ising Model or the study of the transition in finite dimensions for the Spin Glass model in an external field, emphasizing the differences in the results with respect to those obtained by standard perturbative expansion.