Shattering in the Ising p-spin glass model

We study the Ising p -spin glass model for large p . We show that for any inverse temperature √ln 2<β <√2ln 2 ln 2 < β < 2 ln 2 and any large p , the model exhibits shattering : w.h.p. as n→ ∞ n → ∞ , there exists exponentially many well-separated clusters such that (a) each cluster has exponentially small Gibbs mass, and (b) the clusters collectively contain all but a vanishing fraction of Gibbs mass. Moreover, these clusters consist of configurations with energy near β β . Range of temperatures for which shattering occurs is within the replica symmetric region. To the best of our knowledge, this is the first shattering result regarding the Ising p -spin glass models. Furthermore, we show that for any γ >0 γ > 0 and any large enough p , the model exhibits an intricate geometrical property known as the multi Overlap Gap Property above the energy value γ √2ln 2 γ 2 ln 2 . Our proofs are elementary, and in particular based on simple applications of the first and the second moment methods.