Ghirlanda-Guerra identities and ultrametricity: An elementary proof in the discrete case

TL;DR

This paper provides a simple, elementary proof that discrete overlap arrays satisfying the Ghirlanda-Guerra identities are ultrametric with probability one, avoiding complex invariance principles.

Contribution

It introduces a new, elementary proof for ultrametricity of discrete overlap arrays satisfying Ghirlanda-Guerra identities, bypassing advanced invariance techniques.

Findings

Discrete overlap arrays are ultrametric with probability one.

Elementary algebraic methods suffice to prove ultrametricity.

The proof simplifies understanding of the Ghirlanda-Guerra identities in the discrete case.

Abstract

In this paper we give another proof of the fact that a random overlap array, which satisfies the Ghirlanda-Guerra identities and whose elements take values in a finite set, is ultrametric with probability one. The new proof bypasses random change of density invariance principles for directing measures of such arrays and, in addition to the Dobvysh-Sudakov representation, is based only on elementary algebraic consequences of the Ghirlanda-Guerra identities.