RR-positivity of nearest neighbor matrices and applications to Gibbs states

We revisit the RR-positivity of nearest neighbor matrices on Z+Z+ and the Gibbs measures on the set of nearest neighbor trajectories on Z+Z+ whose Hamiltonians award either visits to sites or visits to edges. We give conditions that guarantee the RR-positivity or equivalently the existence of the infinite volume Gibbs measure. Moreover, we supply necessary and sufficient conditions for the geometric ergodicity of the associated Markov chain. In this work we generalize and sharpen results obtained in [7] and [11].