Shattering in the Ising Pure $p$-Spin Model

TL;DR

This paper investigates the energy landscape of the Ising pure p-spin model for large p, revealing a complex geometrical structure called the multi Overlap Gap Property and demonstrating shattering phenomena within certain temperature ranges.

Contribution

It provides the first shattering results for the Ising p-spin models, using elementary probabilistic methods and analyzing the model's geometrical properties.

Findings

Presence of multi Overlap Gap Property above certain energy levels

Existence of exponentially many well-separated clusters in the Gibbs measure

Shattering occurs within the replica symmetric temperature region

Abstract

We study the Ising pure $p$-spin model for large $p$. We investigate the landscape of the Hamiltonian of this model. We show that for any $\gamma>0$ and any large enough $p$, the model exhibits an intricate geometrical property known as the multi Overlap Gap Property above the energy value $\gamma\sqrt{2\ln 2}$. We then show that for any inverse temperature $\sqrt{\ln 2}<\beta<\sqrt{2\ln 2}$ and any large $p$, the model exhibits shattering: w.h.p. as $n\to\infty$, there exists exponentially many well-separated 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 $\beta$. 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 models. Our proof is elementary, and in particular based on simple applications of the first and the second moment methods.