Shattering in Pure Spherical Spin Glasses

TL;DR

This paper demonstrates the existence of a shattered phase in pure spherical p-spin glasses, revealing a complex energy landscape with well-separated clusters, and discusses implications for dynamics and sampling algorithms.

Contribution

It introduces the concept of shattering within the replica-symmetric phase of spherical p-spin models and provides quantitative estimates on the Gibbs measure decomposition.

Findings

Existence of a shattered phase with small, well-separated clusters

Long plateau in the two-times correlation function of Langevin dynamics

Shattering implies disorder chaos and failure of stable sampling algorithms

Abstract

We prove the existence of a shattered phase within the replica-symmetric phase of the pure spherical $p$-spin models for $p$ sufficiently large. In this phase, we construct a decomposition of the sphere into well-separated small clusters, each of which has exponentially small Gibbs mass, yet which together carry all but an exponentially small fraction of the Gibbs mass. We achieve this via quantitative estimates on the derivative of the Franz--Parisi potential, which measures the Gibbs mass profile around a typical sample. Corollaries on dynamics are derived, in particular we show the two-times correlation function of stationary Langevin dynamics must have an exponentially long plateau. We further show that shattering implies disorder chaos for the Gibbs measure in the optimal transport sense; this is known to imply failure of sampling algorithms which are stable under perturbation in the same metric.