On absolute continuity of the Gibbs measure under translations (Q1612775)

Let \(\mu\) be a Gibbs measure in \(\Omega={\mathbb R}^{{\mathbb Z}^d}\) and \(H(x)\) be the corresponding Hamiltonian. For \(h\in\Omega\), let \(\mu^h\) be the image of \(\mu\) under the shift \(x\mapsto x+h\). The author shows that, under certain conditions on \(H(\cdot)\), for each \(h\in l^2({\mathbb Z}^d)\) the measures \(\mu^h\) and \(\mu\) are equivalent with Radon-Nikodym derivative given by \[ {d\mu^h\over d\mu}(x)=\exp\{H(x)-H(x-h)\}. \] If \(d=1\) and certain conditions on \(H(\cdot)\) are satisfied, the following dichotomy holds: (1) For each \(h\in l^2({\mathbb Z})\), \(\mu^h\) and \(\mu\) are equivalent; (2) for each \(h\in\Omega\setminus l^2({\mathbb Z})\), \(\mu^h\) and \(\mu\) are mutually singular.