Condensation with two constraints and disorder
Résumé
Motivated by the study of breathers in the disordered Discrete Non Linear Schr\"odinger equation, we study the uniform probability over the intersection of a simplex and an ellipsoid in $n$ dimensions, with quenched disorder in the definition of either the simplex or the ellipsoid. Unless the disorder is too strong, the phase diagram looks like the one without disorder, with a transition separating a fluid phase, where all variables have the same order of magnitude, and a condensed phase, where one variable is much larger than the others. We then show that the condensed phase exhibits "intermediate symmetry breaking": the site hosting the condensate is chosen neither uniformly at random, nor is it fixed by the disorder realization. In particular, the model mimicking the well-studied Discrete Non Linear Schr\"odinger model with frequency disorder shows a very weak symmetry breaking: all variables have a sizable probability to host the condensate (i.e. a breather in a DNLS setting), but its localization is still biased towards variables with a large linear frequency. Throughout the article, our heuristic arguments are complemented with direct Monte Carlo simulations.