Hamiltonians on random walk trajectories

We consider Gibbs measures on the set of paths of nearest-neighbors random walks on Z+. The basic measure is the uniform measure on the set of paths of the simple random walk on Z+ and the Hamiltonian awards each visit to site x ∈ Z+ by an amount αx ∈ R, x ∈ Z+. We give conditions on (αx) that guarantee the existence of the (infinite volume) Gibbs measure. When comparing the measures in Z+ with the corresponding measures in Z, the so-called entropic repulsion appears as a counting effect.