Probabilistic ultraviolet renormalization of the two-dimensional Sine-Gordon model

Abstract

We give a probabilistic approach to the ultraviolet renormalization of the two-dimensional Sine-Gordon model. The model is formally described by the density $\exp(α\int \mathopen{:} \cos(βX) \mathclose{:} )$, where $X$ is a log-correlated Gaussian field. Our construction applies on bounded convex domains for every particle activity $α\in\mathbb R$ and throughout the subcritical range $β^2\in[0,4)$, below the Berezinskii-Kosterlitz-Thouless threshold. We apply the martingale method introduced by Lacoin, Rhodes and Vargas to two dimensions by using refined estimates based on charge conjugation symmetries. Our result extends existing constructions by treating the full subcritical regime for a broad class of covariances.