Ground-state configuration space heterogeneity of random finite-connectivity spin glasses and random constraint satisfaction problems

TL;DR

This paper uncovers a heterogeneity transition in the ground-state configuration space of random finite-connectivity spin glasses and constraint satisfaction problems, revealing the emergence of multiple configuration communities and their impact on system dynamics.

Contribution

It introduces the concept of a heterogeneity transition in ground-state spaces and characterizes it using replica methods, linking structural heterogeneity to dynamical behavior.

Findings

Exponential emergence of configuration communities at critical constraint density.

Non-concave entropy density s(q) indicating heterogeneity.

Configuration communities cause dynamical heterogeneity through entropic trapping.

Abstract

We demonstrate through two case studies, one on the p-spin interaction model and the other on the random K-satisfiability problem, that a heterogeneity transition occurs to the ground-state configuration space of a random finite-connectivity spin glass system at certain critical value of the constraint density. At the transition point, exponentially many configuration communities emerge from the ground-state configuration space, making the entropy density s(q) of configuration-pairs a non-concave function of configuration-pair overlap q. Each configuration community is a collection of relatively similar configurations and it forms a stable thermodynamic phase in the presence of a suitable external field. We calculate s(q) by the replica-symmetric and the first-step replica-symmetry-broken cavity methods, and show by simulations that the configuration space heterogeneity leads to dynamical heterogeneity of particle diffusion processes because of the entropic trapping effect of configuration communities. This work clarifies the fine structure of the ground-state configuration space of random spin glass models, it also sheds light on the glassy behavior of hard-sphere colloidal systems at relatively high particle volume fraction.