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On the Replica Symmetric Solution in General Diluted Spin Glasses

Ratul Biswas, Wei-Kuo Chen, Arnab Sen

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 $α$. 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 $αp (p-1)\leq 1$. 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.

On the Replica Symmetric Solution in General Diluted Spin Glasses

Abstract

We present a unifying approach to studying the replica symmetric solution in general diluted spin glass models on random -uniform hypergraphs with sparsity parameter . 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 . 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.

Paper Structure

This paper contains 32 sections, 31 theorems, 325 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Results for models with Ising spins
  4. Structure of the paper
  5. Examples
  6. Constraint satisfaction problems
  7. Perceptron model
  8. K-SAT model
  9. Diluted models with Ising p-spin interactions
  10. Diluted models with general spins
  11. Potts model
  12. XY model
  13. Continuous hardcore model
  14. The distributional operator
  15. Existence and uniqueness of the fixed point
  16. ...and 17 more sections

Key Result

Theorem 1.1

In the regime the following statements hold true.

Theorems & Definitions (58)

  • Theorem 1.1: Free energy
  • Theorem 1.2: Ground state energy
  • Proposition 1.3
  • Proposition 1.4
  • proof
  • Corollary 1.5
  • Proposition 1.6
  • proof
  • Theorem 2.1
  • Corollary 2.2
  • ...and 48 more