Some Probabilistic Properties of General Topological Markov Chains

TL;DR

This paper explores probabilistic properties of topological Markov chains, establishing limit theorems and ergodic results for models related to Dyson potentials, and connecting Ruelle operators with Markov process theory.

Contribution

It introduces new probabilistic limit theorems and ergodic criteria for topological Markov chains with Dyson potentials, extending results to non-compact settings.

Findings

Established a functional central limit theorem for Dyson models.

Proved a Breimann ergodic theorem for equilibrium measures.

Provided a criterion for full support of conformal measures.

Abstract

In this paper, we utilize the framework of Markov processes to attain a more probabilistic perspective on the theory of transfer operators. In doing so, we establish a functional central limit theorem (FLCT) for an $O(N)$ model associated with Dyson potential on the one-dimensional lattice. We also proof a FLCT on a non-compact alphabet setting, for a model associated with a Dyson type potential on the one-dimensional lattice. A Breimann ergodic theorem for equilibrium measures arising from the Ruelle Operator Formalism is proved. Furthermore, we obtain a qualitative criterion (strong transitivity) to determine when the conformal measure has full support. As an application, we show how to connect the Ruelle operator framework with the perspective of the Hopf theory of Markov Processes.