Log-Sobolev inequality for the $\varphi^4_2$ and $\varphi^4_3$ measures

Understand

The continuum $\varphi^4_2$ and $\varphi^4_3$ 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 $\varphi^4_2$ and $\varphi^4_3$ 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 $\varphi^4_2$ and $\varphi^4_3$ measures obtained using skeleton inequalities.