Résumé (EN)

The Sine-Gordon model is obtained by tilting the law of a log-correlated Gaussian field X defined on a subset of R d by the exponential of its cosine, namely exp(α ∫ cos(βX)). It is an important model in quantum field theory or in statistic physics like in the study of log-gases. In spite of its relatively simple definition, the model has a very rich phenomenology. While the integral ∫ cos(βX) can properly be defined when β 2 < d using the standard Wick normalisation of cos(βX), a more involved renormalization procedure is needed when β 2 ∈ [d, 2d). In particular it exhibits a countable sequence of phase transition accumulating to the left of β = √ 2d, each transitions corresponding to the addition of an extra term in the renormalization scheme. The final threshold β = √ 2 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 β = √ 2. 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(βX) 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) ∫ cos(βX). To control all terms produced by the induction proceedure, we prove a refinement of classical electrostatic inequalities, which allows to bound the energy of configurations in terms of the Wasserstein distance between + and − charges.