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We study the Ising pure $p$-spin model for large $p$. We investigate thelandscape of the Hamiltonian of this model. We show that for any $\gamma>0$ andany large enough $p$, the model exhibits an intricate geometrical propertyknown as the multi Overlap Gap Property above the energy value$\gamma\sqrt{2\ln 2}$. We then show that for any inverse temperature $\sqrt{\ln2}<\beta<\sqrt{2\ln 2}$ and any large $p$, the model exhibits shattering:w.h.p. as $n\to\infty$, there exists exponentially many well-separated clusterssuch that (a) each cluster has exponentially small Gibbs mass, and (b) theclusters collectively contain all but a vanishing fraction of Gibbs mass.Moreover, these clusters consist of configurations with energy near $\beta$.Range of temperatures for which shattering occurs is within the replicasymmetric region. To the best of our knowledge, this is the first shatteringresult regarding the Ising $p$-spin models. Our proof is elementary, and inparticular based on simple applications of the first and the second momentmethods.

Shattering in the Ising Pure $p$-Spin Model