Approximate Ultrametricity for Random Measures and Applications to Spin Glasses

TL;DR

This paper introduces the concept of Approximate Ultrametricity for random measures, providing conditions for its occurrence and applying it to prove conjectures in mean field spin glasses.

Contribution

It defines Approximate Ultrametricity, characterizes cluster measures, and applies these results to classical spin glass models, confirming key conjectures.

Findings

Cluster measures converge to Ruelle Probability Cascade weights

Provides a sufficient condition for approximate ultrametricity in infinite-dimensional spaces

Proves two major conjectures in mixed p-spin glasses

Abstract

In this paper, we introduce a notion called "Approximate Ultrametricity" which encapsulates the phenomenology of a sequence of random probability measures having supports that behave like ultrametric spaces insofar as they decompose into nested balls. We provide a sufficient condition for a sequence of random probability measures on the unit ball of an infinite dimensional separable Hilbert space to admit such a decomposition, whose elements we call clusters. We also characterize the laws of the measures of the clusters by showing that they converge in law to the weights of a Ruelle Probability Cascade. These results apply to a large class of classical models in mean field spin glasses. We illustrate the notion of approximate ultrametricity by proving two important conjectures regarding mixed p-spin glasses.