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

Lacoin, Hubert; Rhodes, Rémi; Vargas, Vincent (2022), A probabilistic approach of ultraviolet renormalisation in the boundary Sine-Gordon model, Probability Theory and Related Fields, p. 24. 10.1007/s00440-022-01174-5

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Type
Article accepté pour publication ou publié
Date
2022
Journal name
Probability Theory and Related Fields
Publisher
Springer
Published in
Paris
Pages
24
Publication identifier
10.1007/s00440-022-01174-5
Metadata
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Author(s)
Lacoin, Hubert
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Laboratoire de Probabilités et Modèles Aléatoires [LPMA]
Rhodes, Rémi
Institut de Mathématiques de Marseille [I2M]
Vargas, Vincent
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (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.
Subjects / Keywords
Boundary Sine-Gordon; renormalization; Onsager inequality; charge correlation functions

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