Skip to main content

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.

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

Fig. 1

Notes

  1. In which case the dimension is \(d=2\) and KT transition occurs at \(\beta ^2=2d=4\).

  2. Based on an exact perturbative expansion of the Anderson–Yuval reformulation of the anisotropic Kondo problem, see [9].

  3. 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.

  4. Note added in proof: the formula relating cumulants with martingale brackets (Lemma 3.1) has recently found a large variety of applications, including finance: see [11].

References

  1. 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)

    Article  MathSciNet  MATH  Google Scholar 

  2. Biermé, H., Durieu, O., Wang, Y.: Generalized random fields and Lévy’s continuity theorem on the space of tempered distributions. arXiv:1706.09326

  3. Brydges, D., Kennedy, T.: Mayer expansions and the Hamilton–Jacobi equations. J. Stat. Phys. 48(1–2), 19–49 (1987)

    Article  MathSciNet  Google Scholar 

  4. Chandra, A., Hairer, M., Shen, H.: The dynamical Sine-Gordon model in the full subcritical regime. arXiv:1808.02594

  5. Dimock, J., Hurd, T.R.: Sine-Gordon revisited. Ann. Henri Poincaré 1(3), 499–541 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Falco, P.: Kosterlitz–Thouless transition line for the two dimensional Coulomb gas. Commun. Math. Phys. 312, 559–609 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Article  MATH  Google Scholar 

  9. Fendley, J., Saleur, H.: Exact perturbative solution of the Kondo problem. Phys. Rev. Lett. 75(24), 4492–4495 (1996). arXiv:950610

    Article  Google Scholar 

  10. 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)

    Book  MATH  Google Scholar 

  11. Friz, P.K., Gatheral, J., Radoičić, R.: Cumulants and martingales. arXiv:2002.01448

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fröhlich, J., Spencer, T.: The Kosterlitz–Thouless transition in two-dimensional Abelian spin systems and Coulomb gas. Commun. Math. Phys. 81, 527–602 (1981)

    Article  MathSciNet  Google Scholar 

  14. Hairer, M., Shen, H.: The dynamical Sine-Gordon model. Commun. Math. Phys. 341, 341–933 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jacod, J., Kowalski, E., Nikeghbali, A.: Mod-Gaussian convergence: new limit theorems in probability and number theory. Forum Math. 23(4), 835–873 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Junnila, J., Saksman, E.: Christian Webb: Imaginary multiplicative chaos: moments, regularity and connections to the Ising model. arXiv:1806.02118

  17. Leblé, T., Serfaty, S., Zeitouni, O.: Large deviations for the two-dimensional two-component plasma. Commun. Math. Phys. 350(1), 301–360 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nicolò, F.: On the massive sine-Gordon equation in the higher regions of collapse. Commun. Math. Phys. 88(4), 581–600 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Onsager, L.: Electrostatic interaction between molecules. J. Phys. Chem. 43, 189–196 (1939)

    Article  Google Scholar 

  21. Rhodes, R., Vargas, V.: Multidimensional multifractal random measures. Electron. J. Probab. 15, 241–258 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  22. Solovej, J.P.: Some simple counterexamples to the Onsager lemma on metric spaces. Rep. Math. Phys. 28(3), 335–338 (1989)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Hubert Lacoin.

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

$$\begin{aligned} M_t= -\beta \int _{I} \Big ( \int _0^t \sin (\beta X_s(x)) e^{\frac{\beta ^2 }{2}K_s(x,x)}\textrm{d}X_s(x) \Big ) \mu (\textrm{d}x) - \mu (I). \end{aligned}$$
(A.1)

By transforming the product of sines as follows

$$\begin{aligned} 2 \sin (\beta X_1)\sin (\beta X_2)= \cos \left( \beta (X_1-X_2)\right) -\cos \left( \beta ( X_1+X_2)\right) , \end{aligned}$$

The definition in (3.3) and Eq. (4.1) yield

$$\begin{aligned} A^{(2)}_t&= \frac{\beta ^2}{4} \int ^t_0 \int _{I^2} Q_u(x_1,x_2) e^{\frac{\beta ^2}{2}(K_u(x_1,x_1)+K_u(x_2,x_2))} \nonumber \\&\quad \times \left[ \cos \left( \beta (X_u(x_1)-X_u(x_2))\right) -\cos \left( \beta ( X_u(x_1)+X_u(x_2))\right) \right] \mu (\textrm{d}x_1)\mu ( \textrm{d}x_2)\textrm{d}u. \end{aligned}$$
(A.2)

1.2 Computation of \(M^{(2)}\)

Now, we proceed with the computation of \(M^{(2)}\). Recall that for \(u >s\) we have

$$\begin{aligned}{} & {} {\mathbb {E}}\big [ \cos (\beta (X_u(x)-X_u(y))) e^{\frac{\beta ^2}{2} (K_u(x,x) +K_u(y,y))} | {\mathcal {F}}_s \big ]\\{} & {} \quad =\cos (\beta (X_s(x)-X_s(y))) e^{\beta ^2 \int _s^u Q_v(x,y) dv } e^{\frac{\beta ^2}{2} (K_s(x,x)+K_s(y,y))} \end{aligned}$$

and

$$\begin{aligned}{} & {} {\mathbb {E}}\big [ \cos (\beta (X_u(x)+X_u(y))) e^{\frac{\beta ^2}{2} (K_u(x,x) +K_u(y,y))} | {\mathcal {F}}_s \big ] \\{} & {} \quad =\cos (\beta (X_s(x)+X_s(y))) e^{-\beta ^2 \int _s^u Q_v(x,y) dv } e^{\frac{\beta ^2}{2} (K_s(x,x) +K_s(y,y))} \end{aligned}$$

and therefore we get

$$\begin{aligned} \frac{1}{2}{\mathbb {E}}[ \langle M \rangle _t | {\mathcal {F}}_s ]&=\frac{\beta ^2}{4} \int _{{\mathcal {O}}^2} \cos (\beta (X_s(x)-X_s(y))) e^{\frac{\beta ^2}{2} (K_s(x,x) + K_s(y,y))} \\&\quad \left( \int _s^t Q_u(x,y) e^{\beta ^2 \int _s^u Q_v(x,y) dv } du \right) dx dy \\&\quad -\frac{\beta ^2}{4} \int _{{\mathcal {O}}^2} \cos (\beta (X_s(x)+X_s(y))) e^{\frac{\beta ^2}{2} (K_s(x,x) + K_s(y,y))}\\&\quad \left( \int _s^t Q_u(x,y) e^{-\beta ^2 \int _s^u Q_v(x,y) dv } du \right) dx dy \\&\quad +L_s \\&= \frac{1}{4} \int _{{\mathcal {O}}^2} \cos (\beta (X_s(x)-X_s(y)))\\&\quad e^{\frac{\beta ^2}{2} (K_s(x,x) + K_s(y,y))} (e^{\beta ^2 \int _s^t Q_v(x,y) dv } -1) dx dy \\&\quad +\frac{1}{4} \int _{{\mathcal {O}}^2} \cos (\beta (X_s(x)+X_s(y))) \\&\quad e^{\frac{\beta ^2}{2} (K_s(x,x) + K_s(y,y))} (e^{-\beta ^2 \int _s^t Q_v(x,y) dv }-1 ) dx dy \\&\quad +L_s \end{aligned}$$

where

$$\begin{aligned} L_s=&= \frac{\beta ^2}{4} \int _0^s \int _{{\mathcal {O}}^2} Q_u(x,y) \cos (\beta (X_u(x)-X_u(y))) e^{\frac{\beta ^2}{2} (K_u(x,x) + K_u(y,y))} du dx dy \\&\quad -\frac{\beta ^2}{4} \int _0^s \int _{{\mathcal {O}}^2} Q_u(x,y) \cos (\beta (X_u(x)+X_u(y))) e^{\frac{\beta ^2}{2} (K_u(x,x) + K_u(y,y))} du dx dy \end{aligned}$$

We have

$$\begin{aligned} M^{(2,t)}_s= \frac{1}{2}{\mathbb {E}}[ \langle M \rangle _t | {\mathcal {F}}_s] -\frac{1}{2}{\mathbb {E}}[ \langle M \rangle _t ] \end{aligned}$$

Using It’s formula, we have

$$\begin{aligned}&d \cos (\beta (X_s(x)-X_s(y))) e^{\frac{\beta ^2}{2} (K_s(x,x) +K_s(y,y))} \left( \int _s^t e^{\beta ^2 \int _s^u Q_v(x,y) dv } du\right) \\&\quad = -\beta \sin (\beta (X_s(x)-X_s(y))) e^{\frac{\beta ^2}{2} (K_s(x,x) + K_s(y,y))} \\&\qquad \left( \int _s^t e^{\beta ^2 \int _s^u Q_v(x,y) dv } du \right) (dX_s(x)-dX_s(y)) + \cdots \end{aligned}$$

where \(\cdots \) is a finite variation term. Hence we get that

$$\begin{aligned}&d \langle M^{(1,t)}, M^{(2,t)} \rangle _s \\&\quad = \frac{\beta ^4}{4} \int _{{\mathcal {O}}^3} \sin (\beta (X_s(x_1))) \sin (\beta (X_s(x_2)-X_s(x_3))) e^{\frac{\beta ^2}{2} (K_s(x_1,x_1)+K_s(x_2,x_2) + K_s(x_3,x_3))} \\&\qquad \times \left( \int _s^t Q_u(x_2,x_3) e^{\beta ^2 \int _s^u Q_v(x_2,x_3) dv } du \right) (Q_s(x_1,x_2)-Q_s(x_1,x_3)) dx_1 dx_2 dx_3 \\&\qquad + \frac{\beta ^4}{4} \int _{{\mathcal {O}}^3} \sin (\beta (X_s(x_1))) \sin (\beta (X_s(x_2)+X_s(x_3))) e^{\frac{\beta ^2}{2} (K_s(x_1,x_1)+K_s(x_2,x_2) + K_s(x_3,x_3)))} \\&\qquad \times \left( \int _s^t Q_u(x_2,x_3) e^{-\beta ^2 \int _s^u Q_v(x_2,x_3) dv } du \right) (Q_s(x_1,x_2)+Q_s(x_1,x_3)) dx_1 dx_2dx_3 \\&\quad = \frac{\beta ^2}{4} \int _{{\mathcal {O}}^3} \sin (\beta (X_s(x_1))) \sin (\beta (X_s(x_2)-X_s(x_3))) e^{\frac{\beta ^2}{2} (K_s(x_1,x_1)+K_s(x_2,x_2) + K_s(x_3,x_3))} \\&\qquad \times ( e^{\beta ^2 \int _s^t Q_v(x_2,x_3) dv }-1 ) (Q_s(x_1,x_2)-Q_s(x_1,x_3)) dx_1 dx_2 dx_3 \\&\qquad - \frac{\beta ^2}{4} \int _{{\mathcal {O}}^3} \sin (\beta (X_s(x_1))) \sin (\beta (X_s(x_2)+X_s(x_3))) e^{\frac{\beta ^2}{2} (K_s(x_1,x_1)+K_s(x_2,x_2) + K_s(x_3,x_3)))} \\&\qquad \times ( e^{-\beta ^2 \int _s^t Q_v(x_2,x_3) dv } -1 ) (Q_s(x_1,x_2)+Q_s(x_1,x_3)) dx_1 dx_2dx_3 \\ \end{aligned}$$

If one tries to bound the second term in the above sum, one gets by using \(|\sin | \;\leqslant \;1\)

$$\begin{aligned}&\frac{\beta ^2}{4} | \int _{{\mathcal {O}}^3} \sin (\beta (X_s(x_1))) \sin (\beta (X_s(x_2)+X_s(x_3))) e^{\frac{\beta ^2}{2} (K_s(x_1,x_1)+K_s(x_2,x_2) + K_s(x_3,x_3)))} \\&\qquad \times ( e^{-\beta ^2 \int _s^t Q_v(x_2,x_3) dv } -1 ) (Q_s(x_1,x_2)+Q_s(x_1,x_3)) dx_1 dx_2dx_3 | \\&\quad \;\leqslant \;\frac{\beta ^2}{4} \int _{{\mathcal {O}}^3} e^{\frac{\beta ^2}{2} (K_s(x_1,x_1)+K_s(x_2,x_2) + K_s(x_3,x_3)))} \\&\qquad \times (Q_s(x_1,x_2)+Q_s(x_1,x_3)) dx_1 dx_2dx_3 \\&\quad \;\leqslant \;C e^{\frac{3 \beta ^2}{2} s} \int _{{\mathcal {O}}^3} (Q_s(x_1,x_2)+Q_s(x_1,x_3)) dx_1 dx_2dx_3 \\&\quad \;\leqslant \;C e^{(\frac{3 \beta ^2}{2}-d) s} \end{aligned}$$

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

$$\begin{aligned}&\big \langle \alpha M^{(1,t)} + \alpha ^2 M^{(2,t)}+\alpha ^3 M^{(3,t)} \big \rangle _t \\&\quad = \alpha ^2 \big \langle M^{(1,t)} \big \rangle _t +2 \alpha ^3 \big \langle M^{(1,t)}, M^{(2,t)} \big \rangle _t+ 2 \alpha ^4 \big \langle M^{(1,t)}, M^{(3,t)}\big \rangle _t \\&\qquad +\alpha ^4 \big \langle M^{(2,t)} \big \rangle _t+ 2\alpha ^5 \big \langle M^{(2,t)}, M^{(3,t)} \big \rangle _t+ \alpha ^6 \big \langle M^{(3,t)}\big \rangle _t \end{aligned}$$

the cumulant decomposition yields

$$\begin{aligned} {\mathbb {E}}\big [e^{\alpha M_t}\big ]&= e^{\alpha |{\mathcal {O}}| + \frac{\alpha ^2}{2}{\mathbb {E}}\big [ \langle M \rangle _t \big ] +\alpha ^3 {\mathbb {E}}\big [ \langle M^{(1)}, M^{(2)} \rangle _t \big ] } {\mathbb {E}}\\&\quad \left[ e^{ \alpha M^{(1,t)}_t+\alpha ^2 M^{(2,t)}_t +\alpha ^3 M^{(3,t)}_t - \frac{1}{2} \langle \alpha M^{(1,t)} + \alpha ^2 M^{(2,t)}+ \alpha ^3 M^{(3,t)} \rangle _t }\right. \\&\quad \left. \times e^{ \frac{\alpha ^4}{2} \langle M^{(2,t)} \rangle _t +\alpha ^4 \langle M^{(1,t)},M^{(3,t)} \rangle _t + \alpha ^5 \langle M^{(2,t)}, M^{(3,t)} \rangle _t+ \frac{\alpha ^6}{2} \langle M^{(3,t)} \rangle _t}\right] \end{aligned}$$

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.

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • 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