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

TL;DR

This paper proves that the continuum $\

Contribution

It establishes a uniform log-Sobolev inequality for $\

Findings

Log-Sobolev inequality holds uniformly in volume and coupling constants.

The proof employs the Polchinski equation and correlation inequalities.

Susceptibility bounds are crucial for the results.

Abstract

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.