Free energy of a diluted spin glass model with quadratic Hamiltonian

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) ≤ 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.

Some Rigorous Results on the Lévy Spin Glass Model

“…In the hope of bridging the gap between the mean-field model and realistic models of short-range interactions, diluted variants of the Sherrington-Kirkpatrick model have been introduced, such as the spin glass model on a Bethe lattice, the Viana-Bray model, and the diluted p-spin models (see [32,36,49]) that possess a bounded average number of spin interactions at any site as opposed to the diverging connectivity in the fully-connected models. For recent progress on the diluted and related models, see [1,5,11,18,19,21,22,25,26,36,43].…”