R-positivity of matrices and Hamiltonians on nearest neighbors trajectories

TL;DR

This paper investigates conditions for R-positivity of matrices and Gibbs measures on nearest neighbors trajectories, providing generalized criteria and establishing recurrence properties of the related Markov chains.

Contribution

It generalizes and refines previous results on R-positivity and Gibbs measures for nearest neighbors models on the positive integers.

Findings

Conditions guaranteeing R-positivity and existence of Gibbs measures.

Geometrical recurrence of the associated Markov chain.

Generalization and sharpening of earlier results.

Abstract

We revisit the $R-$positivity of nearest neighbors matrices on ${\ZZ_+}$ and the Gibbs measures on the set of nearest neighbors trajectories on ${\ZZ_+}$ whose Hamiltonians award either visits to sites a or visits to edges. We give conditions that guarantee the $R-$positivity or equivalently the existence of the infinite volume Gibbs measure, and we show geometrical recurrence of the associated Markov chain. In this work we generalize and sharpen results obtained in [3] and [6].