  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].
