A tomography of the GREM: beyond the REM conjecture

TL;DR

This paper investigates the local energy statistics of the generalized random energy model, revealing that the previously conjectured universal behavior breaks down at high energies proportional to system size, indicating more complex underlying phenomena.

Contribution

It demonstrates the failure of the GREM energy statistics conjecture at high energies and characterizes the resulting complex behavior beyond the REM framework.

Findings

Conjecture holds for low energies but fails at energies proportional to volume.

High-energy behavior exhibits more complex, non-universal statistics.

Provides a detailed analysis of the transition from REM-like to complex energy landscapes.

Abstract

Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model. This was proven in a large class of models for energies that do not grow too fast with the system size. Considering the example of the generalized random energy model, we show that the conjecture breaks down for energies proportional to the volume of the system, and describe the far more complex behavior that then sets in.