Uniqueness and non-uniqueness of chains on half lines

TL;DR

This paper explores the relationship between chains and Gibbs measures on the half-line, revealing new phase transition phenomena and disproving existing uniqueness conjectures in the context of long-range ferromagnetic Ising models.

Contribution

It establishes a correspondence between one-sided and two-sided systems, and demonstrates new phase transitions and the failure of certain uniqueness conditions in half-line models.

Findings

Discovered new chains with phase transitions using Dyson's Gibbsian construction

Showed that square summability is not a necessary condition for chain uniqueness in non-shift-invariant settings

Disproved a Gibbsian conjecture by Kac and Thompson in the half-line context

Abstract

We establish a one-to-one correspondence between one-sided and two-sided regular systems of conditional probabilities on the half-line that preserves the associated chains and Gibbs measures. As an application, we determine uniqueness and non-uniqueness regimes in one-sided versions of ferromagnetic Ising models with long range interactions. Our study shows that the interplay between chain and Gibbsian theories yields more information than that contained within the known theory of each separate framework. In particular: (i) A Gibbsian construction due to Dyson yields a new family of chains with phase transitions; (ii) these transitions show that a square summability uniqueness condition of chains is false in the general non-shift-invariant setting, and (iii) an uniqueness criterion for chains shows that a Gibbsian conjecture due to Kac and Thompson is false in this half-line setting.