# Average Gromov hyperbolicity and the Parisi ansatz

## TL;DR

This paper introduces an average-case version of Gromov hyperbolicity to determine when a space resembles a tree, and applies this to construct hierarchically organized states in spin glass models following the Parisi ultrametricity ansatz.

## Contribution

It defines an average hyperbolicity measure, proves that small average hyperbolicity implies approximate tree embedding, and applies this to spin glass models.

## Key findings

- Average hyperbolicity is bounded above by Gromov hyperbolicity.
- Small average hyperbolicity implies approximate tree embedding.
- Constructs hierarchically organized pure states in spin glasses.

## Abstract

Gromov hyperbolicity of a metric space measures the distance of the space from a perfect tree-like structure. The measure has a "worst-case" aspect to it, in the sense that it detects a region in the space which sees the maximum deviation from tree-like structure. In this article we introduce an "average-case" version of Gromov hyperbolicity, which detects whether the "most of the space", with respect to a given probability measure, looks like a tree. The main result of the paper is that if this average hyperbolicity is small, then the space can be approximately embedded in a tree. The proof uses a weighted version of Szemeredi's regularity lemma from graph theory. The result applies to Gromov hyperbolic spaces as well, since average hyperbolicity is bounded above by Gromov hyperbolicity. As an application, we give a construction of hierarchically organized pure states in any model of a spin glass that satisfies the Parisi ultrametricity ansatz.