The Parisi ultrametricity conjecture

TL;DR

This paper proves that measures satisfying the Ghirlanda-Guerra identities are ultrametric, confirming the Parisi ultrametricity conjecture for mean-field spin glass models like SK and p-spin models.

Contribution

It establishes the ultrametricity of the support of certain random measures, resolving a key conjecture in spin glass theory.

Findings

Support of measures is ultrametric with probability one

Confirms Parisi ultrametricity conjecture in mean-field models

Ghirlanda-Guerra identities imply ultrametricity

Abstract

In this paper we prove that the support of a random measure on the unit ball of a separable Hilbert space that satisfies the Ghirlanda-Guerra identities must be ultrametric with probability one. This implies the Parisi ultrametricity conjecture in mean-field spin glass models, such as the Sherrington-Kirkpatrick and mixed $p$-spin models, for which Gibbs' measures are known to satisfy the Ghirlanda-Guerra identities in the thermodynamic limit.