Hamiltonians on random walk trajectories
The Gibbs states on the set of those elements \(w\in \mathbb Z_+^{\mathbb Z}\) are studied for which \(| w_i-w_{i+1}| =1\). The considered Hamiltonians are characterized by pair interaction potentials \(\Phi ^\alpha_{\{i,i+1\}}(w)=\alpha (w_{i+1})\) and \(\Phi ^{b,c}_{\{i,i+1\}}(w)= b(w_i)(w_{i+1}-w_i)^++c(w_{i+1})(w_{i+1}-w_i)^-\) for fixed sequences \(\alpha , b, c\) on \(\mathbb Z_+\). The main results give necessary and sufficient conditions on the sequences \(\alpha \), or \(b\) and \(c\), respectively, such that there is a translation invariant Gibbs state corresponding to the interactions \(\Phi ^\alpha \) or \(\Phi ^{b,c}\) provided that \(\alpha , b, c\) are ultimately constant sequences. The key tool is an eigenvalue problem for special infinite matrices corresponding to sequences \(b\) and \(c\). A comparison with symmetric models on \(\mathbb Z^{\mathbb Z}\) goes to an explanation of the phenomenon of entropic repulsion for the studied ``solid on solid models''. The relation to stationary states of an infinite particle system is also discussed.
scientific article; zbMATH DE number 1364514
scientific article