General properties of overlap probability distributions in disordered spin systems. Toward Parisi ultrametricity

TL;DR

This paper proves a general property of overlap distributions in disordered Ising spin systems, showing a specific probabilistic structure of overlaps among replicas in the thermodynamic limit, which supports the Parisi ultrametricity hypothesis.

Contribution

It establishes a universal property of overlap distributions in disordered spin systems, advancing understanding of ultrametricity in the thermodynamic limit.

Findings

Overlap distributions follow a specific probabilistic rule in the thermodynamic limit.

The probability that a new replica's overlap is independent or identical to existing overlaps is 1/s.

Supports the Parisi ultrametricity conjecture in disordered spin systems.

Abstract

For a very general class of probability distributions in disordered Ising spin systems, in the thermodynamical limit, we prove the following property for overlaps among real replicas. Consider the overlaps among s replicas. Add one replica s+1. Then, the overlap q(a,s+1) between one of the first s replicas, let us say a, and the added s+1 is either independent of the former ones, or it is identical to one of the overlaps q(a,b), with b running among the first s replicas, excluding a. Each of these cases has equal probability 1/s.