A connection between the Ghirlanda--Guerra identities and ultrametricity

TL;DR

This paper demonstrates that for a certain class of infinite random matrices, the Ghirlanda--Guerra identities guarantee an ultrametric structure, linking two important concepts in probability theory.

Contribution

It establishes a novel connection showing that Ghirlanda--Guerra identities imply ultrametricity under specific conditions for infinite random matrices.

Findings

Ghirlanda--Guerra identities imply ultrametricity

Finite value condition is crucial for the implication

Results apply to symmetric positive definite weakly exchangeable matrices

Abstract

We consider a symmetric positive definite weakly exchangeable infinite random matrix and show that, under the technical condition that its elements take a finite number of values, the Ghirlanda--Guerra identities imply ultrametricity.