On a nonhierarchical version of the Generalized Random Energy Model. II. Ultrametricity

TL;DR

This paper investigates the conditions under which ultrametricity holds in nonhierarchical Generalized Random Energy Models, showing it requires specific nondegeneracy conditions on the Hamiltonian.

Contribution

It establishes the precise nondegeneracy conditions necessary for ultrametricity in nonhierarchical GREM variants.

Findings

Ultrametricity holds only under certain nondegeneracy conditions.

Nonhierarchical GREM models require specific Hamiltonian properties for ultrametricity.

Provides theoretical criteria for ultrametricity in complex energy landscapes.

Abstract

We study the Gibbs measure of the nonhierarchical versions of the Generalized Random Energy Models introduced in previous work. We prove that the ultrametricity holds only provided some nondegeneracy conditions on the hamiltonian are met.