The Hierarchical Random Energy Model

TL;DR

This paper introduces a hierarchical random energy model with non-mean field interactions, providing evidence for a spin glass transition and revealing non-analytic behavior at the transition point through analytical and numerical methods.

Contribution

It presents a novel hierarchical random energy model and demonstrates a spin glass transition with non-analytic free energy behavior, extending understanding beyond mean field models.

Findings

Evidence of a spin glass condensation transition

Non-analytic high temperature free-energy at transition

Model differs from mean field predictions

Abstract

We introduce a Random Energy Model on a hierarchical lattice where the interaction strength between variables is a decreasing function of their mutual hierarchical distance, making it a non-mean field model. Through small coupling series expansion and a direct numerical solution of the model, we provide evidence for a spin glass condensation transition similar to the one occuring in the usual mean field Random Energy Model. At variance with mean field, the high temperature branch of the free-energy is non-analytic at the transition point.