On the Replica Symmetric Solution in General Diluted Spin Glasses

TL;DR

This paper unifies the understanding of the replica symmetric solution in diluted spin glasses, identifying key regimes where the free energy can be explicitly characterized, and extends these results to various models and problems.

Contribution

It introduces a unifying approach to analyze replica symmetry in diluted spin glasses, revealing explicit free energy representations in key regimes and applying these to multiple models.

Findings

Existence of two key regimes with explicit free energy formulas.

In the high temperature regime, free energy depends on temperature and sparsity.

In the subcritical regime, free energy is independent of temperature.

Abstract

We present a unifying approach to studying the replica symmetric solution in general diluted spin glass models on random $p$-uniform hypergraphs with sparsity parameter $\alpha$. Our result shows that there exist two key regimes in which the model exhibits replica symmetry and the free energy can be explicitly represented as the evaluation of an energy functional at the unique fixed point of a recursive distributional equation. One is called the high temperature regime, where the temperature and the sparsity parameter are essentially inversely proportional to each other; the other is the subcritical regime defined as $\alpha p (p-1)\leq 1$. In particular, the fact that the second regime is independent of the temperature parameter further allows us to deduce an analogous representation of the ground state energy in the subcritical regime. Along the way, we revisit several well-known formulas and also derive new ones for the free and ground state energies in the constraint satisfaction problem, Potts model, XY model, and continuous hardcore model.