Concentration of Multi-overlaps for Random Dilute Ferromagnetic Spin Models

We consider mean field ferromagnetic spin models on dilute random graphs and prove that, with suitable one-body infinitesimal perturbations added to the Hamiltonian, the multi-overlaps concentrate for all temperatures, both with respect to the thermal Gibbs average and the quenched randomness. Results of this nature have been known only for the lowest order overlaps, at high temperature or on the Nishimori line. Here we treat all multi-overlaps by a non-trivial application of Griffiths–Kelly–Sherman correlation inequalities. Our results apply in particular to the pure and mixed p-spin ferromagnets on random dilute Erdoes–Renyi hypergraphs. On physical grounds one expects that multi-overlap concentration is an important ingredient for the validity of the cavity (or replica-symmetric) formula for the pressure of mean field models. However rigorously establishing this formula for the p-spin ferromagnet on a random dilute hypergraph remains an open problem.