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 ln2<β<2ln2 and any large p, the model exhibits shattering: w.h.p. as 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. 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 and any large enough p, the model exhibits an intricate geometrical property known as the multi Overlap Gap Property above the energy value γ2ln2. Our proofs are elementary, and in particular based on simple applications of the first and the second moment methods.