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.
This is a preview of subscription content, access via your institution.

Notes
In which case the dimension is \(d=2\) and KT transition occurs at \(\beta ^2=2d=4\).
Based on an exact perturbative expansion of the Anderson–Yuval reformulation of the anisotropic Kondo problem, see [9].
We consider here the space of tempered distributions as the topological dual of the Schwartz space of smooth functions with fast decay at infinity equipped with the usual semi-norms. We do not recall this here but the reader can refer to [2] for a reminder of the topological setup as well as a proof of the Lévy continuity theorem on the space of tempered distributions.
References
Benfatto, G., Gallavotti, G., Nicolò, F.: On the massive Sine-Gordon equation in the first few regions of collapse. Commun. Math. Phys. 83(3), 387–410 (1982)
Biermé, H., Durieu, O., Wang, Y.: Generalized random fields and Lévy’s continuity theorem on the space of tempered distributions. arXiv:1706.09326
Brydges, D., Kennedy, T.: Mayer expansions and the Hamilton–Jacobi equations. J. Stat. Phys. 48(1–2), 19–49 (1987)
Chandra, A., Hairer, M., Shen, H.: The dynamical Sine-Gordon model in the full subcritical regime. arXiv:1808.02594
Dimock, J., Hurd, T.R.: Sine-Gordon revisited. Ann. Henri Poincaré 1(3), 499–541 (2000)
Falco, P.: Kosterlitz–Thouless transition line for the two dimensional Coulomb gas. Commun. Math. Phys. 312, 559–609 (2012)
Fateev, V., Lukyanov, S., Zamolodchikov, A., Zamolodchikov, A.: Expectation values of boundary fields in the boundary sine-Gordon model. Phys. Lett. B 406, 83–88 (1997)
Fendley, J., Lesage, F., Saleur, H.: Solving 1d plasmas and 2d boundary problems using Jack polynomials and functional relations. J. Stat. Phys. 79(5–6), 799–819 (1995)
Fendley, J., Saleur, H.: Exact perturbative solution of the Kondo problem. Phys. Rev. Lett. 75(24), 4492–4495 (1996). arXiv:950610
Feray, V., Meliot, P.-L., Nikeghbali, A.: Mod-\(\phi \) Convergence Normality Zones and Precise Deviations. Springer Briefs in Probability and Mathematical Statistics, Springer, Berlin (2016)
Friz, P.K., Gatheral, J., Radoičić, R.: Cumulants and martingales. arXiv:2002.01448
Fröhlich, J.: Classical and quantum statistical mechanics in one and two dimensions: two component Yukawa and Coulomb systems. Commun. Math. Phys. 47, 233–268 (1976)
Fröhlich, J., Spencer, T.: The Kosterlitz–Thouless transition in two-dimensional Abelian spin systems and Coulomb gas. Commun. Math. Phys. 81, 527–602 (1981)
Hairer, M., Shen, H.: The dynamical Sine-Gordon model. Commun. Math. Phys. 341, 341–933 (2016)
Jacod, J., Kowalski, E., Nikeghbali, A.: Mod-Gaussian convergence: new limit theorems in probability and number theory. Forum Math. 23(4), 835–873 (2011)
Junnila, J., Saksman, E.: Christian Webb: Imaginary multiplicative chaos: moments, regularity and connections to the Ising model. arXiv:1806.02118
Leblé, T., Serfaty, S., Zeitouni, O.: Large deviations for the two-dimensional two-component plasma. Commun. Math. Phys. 350(1), 301–360 (2017)
Nicolò, F.: On the massive sine-Gordon equation in the higher regions of collapse. Commun. Math. Phys. 88(4), 581–600 (1983)
Nicolò, F., Renn, J., Steinmann, A.: On the massive sine-Gordon equation in all regions of collapse. Commun. Math. Phys. 105(2), 291–326 (1986)
Onsager, L.: Electrostatic interaction between molecules. J. Phys. Chem. 43, 189–196 (1939)
Rhodes, R., Vargas, V.: Multidimensional multifractal random measures. Electron. J. Probab. 15, 241–258 (2010)
Solovej, J.P.: Some simple counterexamples to the Onsager lemma on metric spaces. Rep. Math. Phys. 28(3), 335–338 (1989)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
H. Lacoin: Supported by FAPERj (Grant JCNE) and CPNq (Grant Universal and productivity grant).
R. Rhodes and V. Vargas: Partially supported by Grant ANR-15-CE40-0013 Liouville.
Appendix A. Explicit expression for the first two martingale brackets
Appendix A. Explicit expression for the first two martingale brackets
We compute here the first two brackets explicitly; we make no assumptions on the dimension d and we suppose that \(\beta ^2>d\). We argue that up to the second threshold \(\beta _2^2= \frac{3d}{2}\), the bracket \(\langle M^{(1,t)}, M^{(2,t)} \rangle _t\) is not bounded hence rendering the normalization in this case already quite involved.
1.1 Computation of \(A^{(2)}\)
Recall that we have
By transforming the product of sines as follows
The definition in (3.3) and Eq. (4.1) yield
1.2 Computation of \(M^{(2)}\)
Now, we proceed with the computation of \(M^{(2)}\). Recall that for \(u >s\) we have
and
and therefore we get
where
We have
Using It’s formula, we have
where \(\cdots \) is a finite variation term. Hence we get that
If one tries to bound the second term in the above sum, one gets by using \(|\sin | \;\leqslant \;1\)
Hence, by integration over s the term \(\langle M^{(1,t)}, M^{(2,t)} \rangle _t \) can not be bounded since \(\beta ^2>d\). This shows that as soon as \(\beta ^2>d\), one must go to order three in the renormalisation scheme; since
the cumulant decomposition yields
and one must bound the terms \(\langle M^{(2,t)} \rangle _t, \langle M^{(1,t)},M^{(3,t)} \rangle _t, \langle M^{(2,t)}, M^{(3,t)}\rangle _t, \langle M^{(3,t)} \rangle _t \). Explicit computations of these cases are already rather involved at this point and justify our less explicit approach.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lacoin, H., Rhodes, R. & Vargas, V. A probabilistic approach of ultraviolet renormalization in the boundary Sine-Gordon model. Probab. Theory Relat. Fields 185, 1–40 (2023). https://doi.org/10.1007/s00440-022-01174-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-022-01174-5
Keywords
- Boundary Sine-Gordon
- Renormalization
- Onsager inequality
- Charge correlation functions
Mathematics Subject Classification
- 82B40
- 60F99
- 60H99