Near-optimal shattering in the Ising pure p-spin and rarity of solutions returned by stable algorithms

TL;DR

This paper investigates the structure of solutions in the Ising pure p-spin model, revealing shattering phenomena at certain temperatures and demonstrating that stable algorithms are exponentially unlikely to find typical solutions.

Contribution

It establishes the occurrence of shattering at all inverse temperatures in a specific range and links this to the rarity of solutions found by stable algorithms, highlighting fundamental computational limitations.

Findings

Shattering occurs at all inverse temperatures in the specified range for large p.

A 'soft' overlap gap property is proven, indicating a distance gap in typical solutions.

Stable algorithms are exponentially unlikely to find solutions of typical energy levels.

Abstract

We show that in the Ising pure $p$-spin model of spin glasses, shattering takes place at all inverse temperatures $\beta \in (\sqrt{(2 \log p)/p}, \sqrt{2\log 2})$ when $p$ is sufficiently large as a function of $\beta$. 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.