Skip to main content
SpringerLink
Log in

Exact Renormalization Groups and Transportation of Measures

  • Original Paper
  • Published:
Annales Henri Poincaré Aims and scope Submit manuscript

Abstract

This note provides a new perspective on Polchinski’s exact renormalization group, by explaining how it gives rise, via the multiscale Bakry-Émery criterion, to Lipschitz transport maps between Gaussian free fields and interacting quantum and statistical field theories. Consequently, many functional inequalities can be verified for the latter field theories, going beyond the current known results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Addona, D., Muratori, M., Rossi, M.: On the equivalence of Sobolev norms in Malliavin spaces. J. Funct. Anal. 283(7), 109600 (2022)

    Article  MathSciNet  Google Scholar 

  2. Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 348. Springer, Cham (2014)

    Google Scholar 

  3. Barashkov, N., Gubinelli, M.: A variational method for \(\varphi ^4_3\). Duke Math. J. 169, 3339–3415 (2020)

    MathSciNet  Google Scholar 

  4. Barashkov, N.: A stochastic control approach to Sine Gordon EQFT. arXiv preprint arXiv:2203.06626 (2022)

  5. Barashkov, N., Gubinelli, M.: On the variational method for euclidean quantum fields in infinite volume. arXiv preprint arXiv:2112.05562 (2021)

  6. Bauerschmidt, R., Bodineau, T.: Spectral gap critical exponent for Glauber dynamics of hierarchical spin models. Commun. Math. Phys. 373, 1167–1206 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  7. Bauerschmidt, R., Bodineau, T.: Log-Sobolev inequality for the continuum Sine–Gordon model. Commun. Pure Appl. Math. 74, 2064–2113 (2021)

    Article  MathSciNet  Google Scholar 

  8. Bauerschmidt, R., Dagallier, B.: Log–Sobolev inequality for near critical Ising models. arXiv preprint arXiv:2202.02301

  9. Bauerschmidt, R., Dagallier, B.: Log–Sobolev inequality for the \(\varphi ^{4}_{2}\) and \(\varphi ^{4}_{3}\) measures. Comm. Pure Appl. Math., to appear

  10. Bogachev, V.I.: Gaussian Measures, Mathematical Surveys and Monographs, vol. 62. American Mathematical Society, Providence, RI (1998)

    Book  Google Scholar 

  11. Borell, C.: The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975)

    Article  ADS  MathSciNet  Google Scholar 

  12. Brydges, D.C., Kennedy, T.: Mayer expansions and the Hamilton–Jacobi equation. J. Stat. Phys. 48, 19–49 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  13. Cattiaux, P., Guillin, A.: Semi log-concave Markov diffusions. In: Séminaire de Probabilités XLVI, pp. 231–292. Springer, Cham (2014)

    Chapter  Google Scholar 

  14. Chafaï, D.: Entropies, convexity, and functional inequalities: on \(\varphi \)-entropies and \(\varphi \)-Sobolev inequalities. J. Math. Kyoto Univ. 44, 325–363 (2004)

    MathSciNet  Google Scholar 

  15. Chen, Y., Georgiou, T.T., Pavon, M.: Stochastic control liaisons: richard sinkhorn meets gaspard monge on a Schrödinger bridge. SIAM Rev. 63, 249–313 (2021)

    Article  MathSciNet  Google Scholar 

  16. Chen, Y., Eldan, R.: Localization schemes: a framework for proving mixing bounds for Markov chains. arXiv preprint arXiv:2203.04163 (2022)

  17. Cordero-Erausquin, D.: Some applications of mass transport to Gaussian-type inequalities. Arch. Ration. Mech. Anal. 161, 257–269 (2002)

    Article  MathSciNet  Google Scholar 

  18. Cotler, J., Rezchikov, S.: Renormalization group flow as optimal transport. Physical Review D. 2023 Jul 5;108(2):025003

  19. Dai Pra, P.: A stochastic control approach to reciprocal diffusion processes. Appl. Math. Optim. 23, 313–329 (1991)

    Article  MathSciNet  Google Scholar 

  20. Eldan, R.: Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal. 23, 532–569 (2013)

    Article  MathSciNet  Google Scholar 

  21. Faris, W.G.: Ornstein–Uhlenbeck and renormalization semigroups. Mosc. Math. J. 1(471), 389–405 (2001)

    Article  MathSciNet  Google Scholar 

  22. Föllmer, H.: An entropy approach to the time reversal of diffusion processes. In: Stochastic Differential Systems (Marseille–Luminy, 1984), pp. 156–163. Springer, Berlin (1984)

    Google Scholar 

  23. Föllmer, H.: Time reversal on Wiener space. In: Stochastic Processes-Mathematics and Physics, pp. 119–129. Springer, Berlin (1984)

    Google Scholar 

  24. Jamison, B.: The Markov processes of Schrödinger. Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 323–331 (1975)

    Article  Google Scholar 

  25. Kim, Y.-H., Milman, E.: A generalization of Caffarelli’s contraction theorem via (reverse) heat flow. Math. Ann. 354, 827–862 (2012)

    Article  MathSciNet  Google Scholar 

  26. Klartag, B., Putterman, E.: Spectral monotonicity under Gaussian convolution. Ann. Fac. Sci. Toulouse Math, to appear

  27. Ledoux, M.: Isoperimetry and Gaussian analysis. In: Lectures on Probability Theory and Statistics, pp. 165–294. Springer, Berlin (1994)

    Google Scholar 

  28. Ledoux, M.: The Concentration of Measure phenomenon, Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence, RI (2001)

    Google Scholar 

  29. Lehec, J.: Representation formula for the entropy and functional inequalities. Ann. Inst. Henri Poincaré Probab. Stat. 49, 885–899 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  30. Léonard, C.: A survey of the Schrödinger problem and some of its connections with optimal transport. Discret. Contin. Dyn. Syst. 34, 1533–1574 (2014)

    Article  Google Scholar 

  31. Mikulincer, D., Shenfeld, Y.: The Brownian transport map. arXiv preprint arXiv:2111.11521 (2021)

  32. Mikulincer, D., Shenfeld, Y.: On the Lipschitz properties of transportation along heat flows, GAFA Seminar Notes. arXiv preprint arXiv:2201.01382

  33. Milman, E.: Spectral estimates, contractions and hypercontractivity. J. Spectr. Theory 8, 669–714 (2018)

    Article  MathSciNet  Google Scholar 

  34. Neeman, J.: Lipschitz changes of variables via heat flow. arXiv preprint arXiv:2201.03403 (2022)

  35. Otto, F., Villani, C.: Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J. Funct. Anal. 173, 361–400 (2000)

    Article  MathSciNet  Google Scholar 

  36. Polchinski, J.: Renormalization and effective Lagrangians. Nucl. Phys. B 231, 269–295 (1984)

    Article  ADS  Google Scholar 

  37. Tanana, A.: Comparison of transport map generated by heat flow interpolation and the optimal transport Brenier map. Commun. Contemp. Math. 23(7), 2050025 (2021)

    Article  MathSciNet  Google Scholar 

  38. Villani, C.: Topics in Optimal Transportation, Graduate Studies in Mathematics, vol. 58. American Mathematical Society, Providence, RI (2003)

    Google Scholar 

Download references

Acknowledgements

Many thanks to Roland Bauerschmidt for interesting conversations on the topic of this paper as well as for providing feedback on this work. I also thank Dan Mikulincer (for telling me about [5]), Max Raginsky for careful comments that improved this manuscript, and the anonymous referee for helpful suggestions. This material is based upon work supported by the National Science Foundation under Award Number 2002022. No data is associated with this manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Yair Shenfeld

Authors
  1. Yair Shenfeld
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yair Shenfeld.

Additional information

Communicated by Christian Maes.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Shenfeld, Y. Exact Renormalization Groups and Transportation of Measures. Ann. Henri Poincaré 25, 1897–1910 (2024). https://doi.org/10.1007/s00023-023-01351-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00023-023-01351-9

Search

Navigation