Log-Sobolev inequality for the φ⁴₂ and φ⁴₃ measures

The continuum (Formula presented.) and (Formula presented.) 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 (Formula presented.) and (Formula presented.) 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 (Formula presented.) and (Formula presented.) measures obtained using skeleton inequalities.