Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms
Ahmed El Alaoui
TL;DR
This work analyzes the Ising pure $p$-spin model to rigorously establish shattering of the Gibbs measure in a broad temperature window, providing a soft overlap gap property as the core mechanism. By studying the local landscape via a planted model and contiguity, the authors construct a shattering decomposition into well-separated, small-diameter clusters that collectively carry nearly all Gibbs mass. They further show that any stable (Lipschitz) algorithm targeting a typical-energy point must operate inside an exponentially small exceptional region, linking geometric glassiness to algorithmic hardness. The results extend to mixtures and the spherical case, clarifying the dynamical transition’s role and highlighting limitations of stable sampling and optimization procedures in the shattered phase.
Abstract
We show that in the Ising pure $p$-spin model of spin glasses, shattering takes place at all inverse temperatures $β\in (\sqrt{(2 \log p)/p}, \sqrt{2\log 2})$ when $p$ is sufficiently large as a function of $β$. Of special interest is the lower boundary of this interval which matches the large $p$ asymptotics of the inverse temperature marking the hypothetical dynamical transition predicted in statistical physics. We show this as a consequence of a `soft' version of the overlap gap property which asserts the existence of a distance gap of points of typical energy from a typical sample from the Gibbs measure. We further show that this latter property implies that stable algorithms seeking to return a point of at least typical energy are confined to an exponentially rare subset of that super-level set, provided that their success probability is not vanishingly small.