Strong replica symmetry for high-dimensional disordered log-concave Gibbs measures. (arXiv:2009.12939v3 [math.PR] UPDATED)

We consider a generic class of log-concave, possibly random, (Gibbs)
measures. We prove the concentration of an infinite family of order parameters
called multioverlaps. Because they completely parametrise the quenched Gibbs
measure of the system, this implies a simple representation of the asymptotic
Gibbs measures, as well as the decoupling of the variables in a strong sense.
These results may prove themselves useful in several contexts. In particular in
machine learning and high-dimensional inference, log-concave measures appear in
convex empirical …

arxiv math pr