Log‐Sobolev inequality for the φ24$\varphi ^4_2$ and φ34$\varphi ^4_3$ measures

Abstract

AbstractThe continuum and measures are shown to satisfy a log‐Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to all coupling constants in any finite volume, and uniformly in the volume in the entire high temperature phases of the and  models.The proof uses a general criterion for the log‐Sobolev inequality in terms of the Polchinski (renormalisation group) equation, a recently proved remarkable correlation inequality for Ising models with general external fields, the Perron–Frobenius theorem, and bounds on the susceptibilities of the and measures obtained using skeleton inequalities.