A probabilistic approach of ultraviolet renormalization in the boundary Sine-Gordon model

Abstract

The Sine-Gordon model is obtained by tilting the law of a log-correlated Gaussian field X defined on a subset of \({\mathbb {R}}^d\) by the exponential of its cosine, namely \(\exp (\alpha \smallint \cos (\beta X))\). It has gathered significant attention due to its importance in quantum field theory and to its connection with the study of log-gases in statistical mechanics. In spite of its relatively simple definition, the model has a very rich phenomenology. While the integral \(\smallint \cos (\beta X)\) can be defined properly when \(\beta ^2<d\) using the standard Wick normalization of \(\cos (\beta X)\), a more involved renormalization procedure is needed when \(\beta ^2\in [d,2d)\). In particular it exhibits a countable sequence of phase transitions accumulating to the left of \(\beta =\sqrt{2d}\), each transition corresponding to the addition of an extra term in the renormalization scheme. The final threshold \(\beta =\sqrt{2d}\) corresponds to the Kosterlitz–Thouless (KT) phase transition of the \(\log \)-gas. In this paper, we present a novel probabilistic approach to renormalization of the two-dimensional boundary (or 1-dimensional) Sine-Gordon model up to the KT threshold \(\beta =\sqrt{2d}\). The purpose of this approach is to propose a simple and flexible method to treat this problem which, unlike the existing renormalization group techniques, does not rely on translation invariance for the covariance kernel of X or the reference measure along which \(\cos (\beta X)\) is integrated. To this purpose we establish by induction a general formula for the cumulants of a random variable defined on a filtered probability space expressed in terms of brackets of a family of martingales; to the best of our knowledge, the recursion formula is new and might have other applications. We apply this formula to study the cumulants of (approximations of) \(\smallint \cos (\beta X)\). To control all terms produced by the induction procedure, we prove a refinement of classical electrostatic inequalities, which allows us to bound the energy of configurations in terms of the Wasserstein distance between \(+\) and − charges.