Self-Sustained Clusters as Drivers of Computational Hardness in $p$-spin Models

TL;DR

This paper explores the microscopic structures called self-sustained clusters in the $p$-spin model, revealing their role in the system's macroscopic phases and computational hardness across different temperatures.

Contribution

It introduces the concept of self-sustained clusters in the $p$-spin model and analyzes their entropy and properties, linking microscopic configurations to macroscopic phases.

Findings

Self-sustained clusters exist with measurable entropy across temperatures.

In-cluster fields dominate out-cluster fields, indicating stable configurations.

Observations in finite lattices support the microscopic findings.

Abstract

While macroscopic properties of spin glasses have been thoroughly investigated, their manifestation in the corresponding microscopic configurations is much less understood. Cases where both descriptions have been provided, such as constraint satisfaction problems, are limited to their ground state properties. To identify the emerging microscopic structures with macroscopic phases at different temperatures, we study the $p$-spin model with $p\!=\!3$. We investigate the properties of self-sustained clusters, defined as variable sets where in-cluster induced fields dominate over the field induced by out-cluster spins, giving rise to stable configurations with respect to fluctuations. We compute the entropy of self-sustained clusters as a function of temperature and their sizes. In-cluster fields properties and the difference between in-cluster and out-cluster fields support the observation of slow-evolving spins in spin models. The findings are corroborated by observations in finite dimensional lattices at low temperatures.