Skip to main content
Log in

Lattice Gauge Theory and a Random-Medium Ising Model

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

We study linearization of lattice gauge theory. Linearized theory approximates lattice gauge theory in the same manner as the loop O(n)-model approximates the spin O(n)-model. Under mild assumptions, we show that the expectation of an observable in linearized Abelian gauge theory coincides with the expectation in the Ising model with random edge-weights. We find a similar relation between Yang-Mills theory and 4-state Potts model. For the latter, we introduce a new observable.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Math. Acad. Sci. Paris 333(3), 239–244 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  2. Khristoforov, M., Smirnov, S.: Percolation and O(1) loop model, preprint (2021). arXiv:2111.15612

  3. Duminil-Copin, H., Peled, R., Samotij, W., Spinka, Y.: Exponential decay of loop lengths in the loop O(n) model with large n. arXiv:1412.8326v3

  4. Glimm, J., Jaffe, A.: Quantum Physics: A Functional Integral Point of View. Springer, Berlin (2012)

    MATH  Google Scholar 

  5. Gattringer, C., Lang, C.B.: Quantum Chromodynamics on the Lattice: An Introductory Presentation. Lecture Notes in Physics, vol. 788. Springer, Berlin (2010)

    Book  Google Scholar 

  6. Migdal, A.A.: Recursion equations in gauge theories. Sov. Phys. JETP 42, 413 (1975)

    ADS  Google Scholar 

  7. Peled, R., Spinka, Y.: Lecture notes on the spin and loop O(n) models, preprint (2016). arXiv:1708.00058

  8. Guth, A.H.: Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory. Phys. Rev. D 21, 2291–2307 (1980)

    Article  ADS  MathSciNet  Google Scholar 

  9. Maldacena, J.: The symmetry and simplicity of the laws of physics and the Higgs boson. Eur. J. Phys. 37, 1 (2016). arXiv:1410.6753

    Article  Google Scholar 

  10. Balakin, A., Petrova, Yu., Shtern, A., Smirnov, A., Skopenkov, M.: Quarks game, 29th Summer conference of the International mathematical Tournament of towns, (2017). https://www.turgor.ru/lktg/2017/4/4-1en-sol.pdf

  11. Ilinski, K.: Physics of Finance – Gauge Modelling in Non-equilibrium Pricing. Wiley, New York (2001)

    Google Scholar 

  12. Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228, 1590–1630 (2011)

    Article  MathSciNet  Google Scholar 

  13. Bing, R.H.: The geometric topology of 3-manifolds, AMS Colloq. Publ. 40 (1983)

  14. Penrose, R.: Applications of negative dimensional tensors. In: Welsh, D.J.A. (ed.) Combinatorial Mathematics and Its Applications (Proceedings Conference, 1969), pp. 221–244. Academic Press, Oxford (1971)

    Google Scholar 

  15. Ootsuka, T., Tanaka, E., Loginov, E.: Non-associative gauge theory, 2005. arXiv:hep-th/0512349v2

  16. Fedorov, M.: Some aspects of probability distribution for percolation of several fluids on the hexagonal lattice, preprint, (2019). arXiv:1908.11783

  17. Novikov, I.: Percolation of three fluids on a honeycomb lattice, preprint (2019). arXiv:1912.01757

Download references

Acknowledgements

For the latter conjecture, there have been suggested a proof by K. Izyurov and A. Magazinov, as well as interesting generalizations by M. Fedorov and I. Novikov (private communication) [16, 17]. The author is grateful to D. Chelkak, H. Duminil-Copin, M. Khristoforov, S. Melikhov, S. Smirnov for useful discussions.

Funding

The work is supported by Ministry of Science and Higher Education of the Russian Federation, agreement N075-15-2019-1619.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mikhail Skopenkov.

Ethics declarations

Conflict of interest

The author has no relevant financial or non-financial interests to disclose.

Additional information

Dedicated to the last real scientists, brave to face real difficulties, not sweeping them under the rug.

The work is supported by Ministry of Science and Higher Education of the Russian Federation, agreement N075-15-2019-1619.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Skopenkov, M. Lattice Gauge Theory and a Random-Medium Ising Model. Math Phys Anal Geom 25, 18 (2022). https://doi.org/10.1007/s11040-022-09430-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-022-09430-9

Keywords

Mathematics Subject Classification

Navigation