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Concentration of multi-overlaps for random ferromagnetic spin models

We consider 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-Rényi hypergraphs. On physical grounds one expects that multi-overlap concentration directly implies the correctness of the cavity (or replica symmetric) formula for the pressure. The proof of this formula for the general p-spin ferromagnet on a random dilute hypergraph remains an open problem.