Random-energy model: An exactly solvable model of disordered systems

TLDR

The random‑energy model represents the limit of disordered systems where energy level correlations vanish. The paper introduces and solves the random‑energy model of disordered systems. The authors analytically solve the model, compute thermodynamic corrections, chart its phase diagram with ferromagnetic interactions, and analyze the replica method. The model displays a phase transition to a fully frozen low‑temperature phase, constant susceptibility, and results qualitatively match the Sherrington‑Kirkpatrick model, providing lower bounds for ground‑state energies of many spin‑glass systems.

Abstract

A simple model of disordered systems---the random-energy model---is introduced and solved. This model is the limit of a family of disordered models, when the correlations between the energy levels become negligible. The model exhibits a phase transition and the low-temperature phase is completely frozen. The corrections to the thermodynamic limit are discussed in detail. The magnetic properties are studied, and a constant susceptibility is found at low temperature. The phase diagram in the presence of ferromagnetic pair interactions is described. Many results are qualitatively the same as those of the Sherrington-Kirkpatrick model. The problem of using the replica method is analyzed. Lastly, this random-energy model provides lower bounds for the ground-state energy of a large class of spin-glass models.