A-localized states for clock models on trees and their extremal decomposition into glassy states

TL;DR

This paper analyzes the extremal decomposition of Gibbs states in $ ext{Z}_q$ clock models on trees, revealing uncountably many inhomogeneous extremal states at strong coupling, using a novel site/bad site decomposition and reconstruction techniques.

Contribution

It provides a detailed description of the extremal decomposition of $ ext{Z}_q$ clock models on trees, introducing new methods for analyzing inhomogeneous states and their concentration properties.

Findings

$ ext{Z}_q$ clock models exhibit uncountably many extremal inhomogeneous states at strong coupling.

The paper introduces a new site/bad site decomposition for analyzing localization properties.

$ ext{A}$-localization property and multi-site reconstruction are key tools in the analysis.

Abstract

We consider $\mathbb{Z}_q$-valued clock models on a regular tree, for general classes of ferromagnetic nearest neighbor interactions which have a discrete rotational symmetry. It has been proved recently that, at strong enough coupling, families of homogeneous Markov chain Gibbs states $\mu_A$ coexist whose single-site marginals concentrate on $A\subset \mathbb{Z}_q$, and which are not convex combinations of each other [AbHeKuMa24]. In this note, we aim at a description of the extremal decomposition of $\mu_A$ for $|A|\geq 2$ into all extremal Gibbs measures, which may be spatially inhomogeneous. First, we show that in regimes of very strong coupling, $\mu_A$ is not extremal. Moreover, $\mu_A$ possesses a single-site reconstruction property which holds for spin values sent from the origin to infinity, when these initial values are chosen from $A$. As our main result, we show that $\mu_A$ decomposes into uncountably many extremal inhomogeneous states. The proof is based on multi-site reconstruction which allows to derive concentration properties of branch overlaps. Our method is based on a new good site/bad site decomposition adapted to the $A$-localization property, together with a coarse graining argument in local state space.