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

R-positivity of matrices and Hamiltonians on nearest neighbors trajectories