Skip to main content
SpringerLink
Log in

A Renormalisation Group Method. III. Perturbative Analysis

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

This paper is the third in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields, fermion fields, or both. In this paper, we motivate and present a general approach towards second-order perturbative renormalisation, and apply it to a specific supersymmetric field theory which represents the continuous-time weakly self-avoiding walk on \({{{\mathbb {Z}}}^{d}}\). Our focus is on the critical dimension \(d=4\). The results include the derivation of the perturbative flow of the coupling constants, with accompanying estimates on the coefficients in the flow. These are essential results for subsequent application to the 4-dimensional weakly self-avoiding walk, including a proof of existence of logarithmic corrections to their critical scaling. With minor modifications, our results also apply to the 4-dimensional \(n\)-component \(|\varphi |^4\) spin model.

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. Bauerschmidt, R.: A simple method for finite range decomposition of quadratic forms and Gaussian fields. Probab. Theory Relat. Fields 157, 817–845 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bauerschmidt, R., Brydges, D.C., Slade, G.: Critical two-point function of the 4-dimensional weakly self-avoiding walk. Commun. Math. Phys. (2014). arXiv:1403.7268

  3. Bauerschmidt, R., Brydges, D.C., Slade, G.: Logarithmic correction for the susceptibility of the 4-dimensional weakly self-avoiding walk: a renormalisation group analysis. Commun. Math. Phys. (2014). arXiv:1403.7422

  4. Bauerschmidt, R., Brydges, D.C., Slade. G.: Ptsoft: python program for perturbative renormalisation group calculation, Version 1.0 [Software]. Available at http://www.math.ubc.ca/~slade/, (2014)

  5. Bauerschmidt, R., Brydges, D.C., Slade. G.: Structural stability of a dynamical system near a non-hyperbolic fixed point. Annales Henri Poincaré (2014). arXiv:1211.2477

  6. Bauerschmidt, R., Brydges, D.C., Slade, G.: Scaling limits and critical behaviour of the \(4\)-dimensional \(n\)-component \(|\varphi |^4\) spin model. J. Stat. Phys. 157, 692–742 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  7. Brydges, D., Slade, G.: Renormalisation group analysis of weakly self-avoiding walk in dimensions four and higher. In: Bhatia, R., et al. (eds.) Proceedings of the International Congress of Mathematicians. Hyderabad 2010, pp. 2232–2257. World Scientific, Singapore (2011)

  8. Brydges, D.C.: Lectures on the renormalisation group. In: Sheffield, S., Spencer, T. (eds.) Statistical Mechanics, pp. 7–93. American Mathematical Society, Providence, (2009). IAS/Park City Mathematics Series, vol. 16

  9. Brydges, D.C., Guadagni, G., Mitter, P.K.: Finite range decomposition of Gaussian processes. J. Stat. Phys. 115, 415–449 (2004)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  10. Brydges, D.C., Imbrie, J.Z., Slade, G.: Functional integral representations for self-avoiding walk. Probab. Surv. 6, 34–61 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brydges, D.C., Slade, G.: A renormalisation group method. I. Gaussian integration and normed algebras. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1163-z

  12. Brydges, D.C., Slade, G.: A renormalisation group method. II. Approximation by local polynomials. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1164-y

  13. Brydges, D.C., Slade, G.: A renormalisation group method. IV. Stability analysis. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1166-9

  14. Brydges, D.C., Slade, G.: A renormalisation group method. V. A single renormalisation group step. J. Stat. Phys. (to appear). doi:10.1007/s10955-014-1167-8

  15. Dieudonné, J.: Foundations of Modern Analysis. Academic Press, New York (1969)

    MATH  Google Scholar 

  16. Gell-Mann, M., Low, F.E.: Quantum electrodynamics at small distances. Phys. Rev. 95, 1300–1312 (1954)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  17. Glimm, J., Jaffe, A.: Quantum Physics. A Functional Integral Point of View, 2nd edn. Springer, Berlin (1987)

    Google Scholar 

  18. Slade, G., Tomberg, A.: Critical correlation functions for the \(4\)-dimensional weakly self-avoiding walk and \(n\)-component \(|\varphi |^4\) model (2014). arXiv:1412.2668

  19. Wilson, K.G.: The renormalization group and critical phenomena. Rev. Mod. Phys. 55, 583–600 (1983)

    Article  ADS  Google Scholar 

Download references

Acknowledgments

This work was supported in part by NSERC of Canada. RB gratefully acknowledges the support and hospitality of the IAM at the University of Bonn, and of the Department of Mathematics and Statistics at McGill University, where part of this work was done. DB gratefully acknowledges the support and hospitality of the Institute for Advanced Study at Princeton and of Eurandom during part of this work. GS gratefully acknowledges the support and hospitality of the Institut Henri Poincaré, where part of this work was done.

Author information

Author notes

  1. Roland Bauerschmidt

    Present address: School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Roland Bauerschmidt, David C. Brydges & Gordon Slade

Authors
  1. Roland Bauerschmidt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David C. Brydges
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gordon Slade
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David C. Brydges.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Bauerschmidt, R., Brydges, D.C. & Slade, G. A Renormalisation Group Method. III. Perturbative Analysis. J Stat Phys 159, 492–529 (2015). https://doi.org/10.1007/s10955-014-1165-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-014-1165-x

Keywords

Mathematics Subject Classification

Search

Navigation