On the Dimension of the Space of Harmonic Functions on Transitive Shift Spaces

TL;DR

This paper links phase transitions in 1D statistical mechanics to the dimension of harmonic function spaces, extending classical transfer operator theory to low regular potentials and exploring implications for equilibrium states.

Contribution

It extends Ruelle-Perron-Frobenius theory to low regular potentials and develops methods to analyze harmonic functions and equilibrium states in both finite and infinite alphabet settings.

Findings

Established a relation between phase transition and harmonic function space dimension.

Determined support of equilibrium states for low regular potentials including phase transitions.

Proved a version of the Functional Central Limit Theorem without spectral gap assumptions.

Abstract

In this paper, we show a new relation between phase transition in one-dimensional Statistical Mechanics and the multiplicity of the dimension of the space of harmonic functions for an extension of the classical transfer operator. We accomplish this by extending the classical Ruelle-Perron-Frobenius theory to the realm of low regular potentials. This is done by establishing finer properties of the associated conformal measures and thoroughly developing a method to obtain information on the maximal eigenspace of a suitably constructed family of Markov Processes. Our results are valid in the setting of finite and infinite alphabets. Several new applications are given to illustrate the theory. For example, we determine the support of a large class of equilibrium states associated with low regular potentials, including ones allowing phase transition. Additionally, we prove a version of the Functional Central Limit Theorem for equilibrium states. A remarkable aspect of this result is that it does not require the spectral gap property of the associated transfer operator. It is valid for long-range spins systems that might not be positively correlated and for non-local observables.