Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures

We consider a generic class of log-concave, possibly random, (Gibbs) measures. Using a new type of perturbation we prove concentration of an infinite family of order parameters called multioverlaps. These completely parametrise the quenched Gibbs measure of the system, so that their self-averaging behavior implies a simple representation of asymptotic Gibbs measures, as well as decoupling of the variables at hand in a strong sense. Our concentration results may prove themselves useful in several contexts. In particular in machine learning and high-dimensional inference, log-concave measures appear in convex empirical risk minimisation, maximum a-posteriori inference or M-estimation. We believe that our results may be applicable in establishing some type of "replica symmetric formulas" for the free energy, inference or generalisation error in such settings.