An ultrametric state space with a dense discrete overlap distribution: Paperfolding sequences

TL;DR

This paper analyzes paperfolding sequences, revealing their ultrametric structure and a dense, discrete overlap distribution on dyadic rationals, illustrating properties relevant to spin glass models.

Contribution

It computes the Parisi overlap distribution for paperfolding sequences, showing it is a pure point measure supported on the full interval [-1, 1], demonstrating ultrametricity.

Findings

Overlap distribution is discrete and dense on dyadic rationals.

The space exhibits an ultrametric structure.

Supports properties suggested for pure states in spin glasses.

Abstract

We compute the Parisi overlap distribution for paperfolding sequences. It turns out to be discrete, and to live on the dyadic rationals. Hence it is a pure point measure whose support is the full interval [-1; +1]. The space of paperfolding sequences has an ultrametric structure. Our example provides an illustration of some properties which were suggested to occur for pure states in spin glass models.