On the origin of ultrametricity

TL;DR

This paper proves that ultrametricity naturally arises in complex systems with non-trivial overlap distributions, based on simple assumptions like replica equivalence and overlap separability.

Contribution

It establishes a general proof of ultrametricity from basic assumptions, clarifying its origin in complex systems.

Findings

Ultrametricity is proven under broad conditions.

The proof relies on replica equivalence and overlap separability.

Ultrametricity emerges in the infinite volume limit.

Abstract

In this paper we show that in systems where the probability distribution of the the overlap is non trivial in the infinity volume limit, the property of ultrametricity can be proved in general starting from two very simple and natural assumptions: each replica is equivalent to the others (replica equivalence or stochastic stability) and all the mutual information about a pair of equilibrium configurations is encoded in their mutual distance or overlap (separability or overlap equivalence).