# Log-Sobolev inequality for the continuum sine-Gordon model

## TL;DR

This paper develops a multiscale criterion based on the Polchinski equation to establish Log-Sobolev inequalities for the continuum Sine-Gordon model, extending applicability beyond log-concave measures.

## Contribution

It introduces a novel multiscale Bakry--Émery criterion leveraging the Polchinski PDE, applicable to non-log-concave measures like the continuum Sine-Gordon model.

## Key findings

- Proves Log-Sobolev inequalities for the massive continuum Sine-Gordon model with 2<6c
- Establishes asymptotically optimal inequalities for Glauber and Kawasaki dynamics
- Results are independent of recent regularity structures for singular SPDEs

## Abstract

We derive a multiscale generalisation of the Bakry--\'Emery criterion for a measure to satisfy a Log-Sobolev inequality. Our criterion relies on the control of an associated PDE well known in renormalisation theory: the Polchinski equation. It implies the usual Bakry--\'Emery criterion, but we show that it remains effective for measures which are far from log-concave. Indeed, using our criterion, we prove that the massive continuum Sine-Gordon model with $\beta < 6\pi$ satisfies asymptotically optimal Log-Sobolev inequalities for Glauber and Kawasaki dynamics. These dynamics can be seen as singular SPDEs recently constructed via regularity structures, but our results are independent of this theory.