The log-Sobolev inequality for unbounded spin systems (Q1304657)

Proofs of log-Sobolev inequalities for Gibbs measures are abundant in the literature, with varying assumptions on the potentials involved. They are directed towards studying the thermodynamic properties of lattice spin systems such as decay of correlations and uniqueness in the thermodynamic (large volume) limit. In the present note, the authors weaken known sufficient conditions. The essential new assumption is ``convexity at infinity'' for the one-body potential \(\varphi(x)\), meaning that there exists a bounded \(C^\infty\) function \(s(x)\) such that \(\varphi+s\) is strictly convex. This includes cases where \(\varphi(x)=ax^4-bx^2\) \((a>0,\;b>0)\). In their proof, the authors refer to yet unpublished results of N. Yoshida.