SpringerLink
Log in

Limiting Measure of Lee–Yang Zeros for the Cayley Tree

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

This paper is devoted to an in-depth study of the limiting measure of Lee–Yang zeroes for the Ising Model on the Cayley Tree. We build on previous works of Müller-Hartmann and Zittartz (Z Phys B 22:59, 1975), Müller-Hartmann (Z Phys B 27:161–168, 1977), Barata and Marchetti (J Stat Phys 88:231–268, 1997) and Barata and Goldbaum (J Stat Phys 103:857–891, 2001), to determine the support of the limiting measure, prove that the limiting measure is not absolutely continuous with respect to Lebesgue measure, and determine the pointwise dimension of the measure at Lebesgue a.e. point on the unit circle and every temperature. The latter is related to the critical exponents for the phase transitions in the model as one crosses the unit circle at Lebesgue a.e. point, providing a global version of the “phase transition of continuous order” discovered by Müller-Hartmann–Zittartz. The key techniques are from dynamical systems because there is an explicit formula for the Lee–Yang zeros of the finite Cayley Tree of level n in terms of the n-th iterate of an expanding Blaschke Product. A subtlety arises because the conjugacies between Blaschke Products at different parameter values are not absolutely continuous.

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

Access this article

Log in via an institution

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 includes VAT (Germany)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barata J., Goldbaum P.: On the distribution and gap structure of Lee–Yang zero ising model: periodic and aperiodic couplings. J. Stat. Phys. 103, 857–891 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barata J., Marchetti D.: Griffiths’ singularities in diluted ising models on the cayley tree. J. Stat. Phys. 88, 231–268 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Baxter, R.J.: Exactly solved models in statistical mechanics. Academic Press, Inc. (Harcourt Brace Jovanovich, Publishers), London (1989). Reprint of the 1982 original.

  4. Bers L., Royden H.L.: Holomorphic families of injections. Acta Math. 157(3-4), 259–286 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. Biskup M., Borgs C., Chayes J.T., Kleinwaks L.J., Kotecký R.: Partition function zeros at first-order phase transitions: a general analysis. Commun. Math. Phys. 251(1), 79–131 (2004)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  6. Bleher P., Lyubich M.: R. Roeder Lee–Yang zeros for the DHL and 2D rational dynamics, I. Foliation of the physical cylinder. J. Math. Pures Appl. (9) 107(5), 491–590 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cardy John L.: Conformal invariance and the Yang–Lee edge singularity in two dimensions. Phys. Rev. Lett. 54, 1354–1356 (1985)

    Article  ADS  MathSciNet  Google Scholar 

  8. Erchenko, A.: Flexibility of exponents for expanding maps on a circle. Discrete Contin. Dyn. Syst. 39(5), 2325–2342 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fisher M. E.: Yang–Lee edge singularity and \({{{\phi}}^{3}}\) field theory. Phys. Rev. Lett. 40, 1610–1613 (1978)

    Article  ADS  Google Scholar 

  10. Galatolo S., Pollicott M.: Controlling the statistical properties of expanding maps. Nonlinearity 30(7), 2737–2751 (2017)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Ilyashenko Y.S., Kleptsyn V.A., Saltykov P.: Openness of the set of boundary preserving maps of an annulus with intermingled attracting basins. J. Fixed Point Theory Appl. 3(2), 449–463 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ivrii, O.: The geometry of the Weil–Petersson metric in complex dynamics. (2015). To appear in Transactions of the AMS. ar**v preprint ar**v:1503.02590

  13. Kifer Y.: Large deviations in dynamical systems and stochastic processes. Trans. Am. Math. Soc. 321(2), 505–524 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kleptsyn V., Ryzhov D., Minkov S.: Special ergodic theorems and dynamical large deviations. Nonlinearity 25(11), 3189–3196 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Krzy zewski, K. (1977) Some results on expanding map**s, pp. 205–218. Astérisque, No. 50

  16. Lee, T., Yang, C.: Statistical theory of equations of state and phase transitions. I. Theory of condensation. Phys. Rev., 87(3) (1952).

  17. Mañé, R.: The Hausdorff dimension of invariant probabilities of rational maps. In: Dynamical systems, Valparaiso 1986, volume 1331 of Lecture Notes in Mathematics, pp. 86–117. Springer, Berlin (1988)

  18. Manning, A.: A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergod. Theory Dyn. Syst. 1(4), 451–459 (1982), 1981

  19. McMullen C.: Dynamics on the Unit Disk: short geodesics and simple cycles. Comment. Math. Helv. 85(4), 723–749 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Milnor J.: Dynamics in One Complex Variable. (AM160:AM160. Princeton University Press, Princeton (2011)

    Book  Google Scholar 

  21. Milnor J.: Fubini foiled: Katok’s paradoxical example in measure theory. Math. Intell. 19(2), 30–32 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Müller-Hartmann E.: Theory of the Ising model on a Cayley Tree. Z. Phys. B 27, 161–168 (1977)

    Article  ADS  Google Scholar 

  23. Müller-Hartmann, E., Zittartz, J.: Phase transitions of continuous order: Ising model on a Cayley tree. Z Phys. B, 22(59) (1975).

  24. Mussardo G., Bonsignori R., Trombettoni A.: Yang lee zeros of the yang lee model. J. Phys. A: Math. Theor. 50(48), 484003 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Peters, H., Regts, G.: Location of zeros for the partition function of the Ising model on bounded degree graphs. ar**v:1810.01699

  26. Pujals E.R., Robert L., Shub M.: Expanding maps of the circle rerevisited: positive Lyapunov exponents in a rich family. Ergod. Theory Dynam. Syst. 26(6), 1931–1937 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Ruelle D.: Extension of the Lee–Yang circle theorem. Phys. Rev. Lett. 26, 303–304 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  28. Ruelle David: An inequality for the entropy of differentiable maps. Bol. Soc. Brasil. Math. 9(1), 83–87 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sacksteder, R.: The measures invariant under an expanding map. Lect. Notes Math. 392, 179–194 (1974)

  30. Shub M.: Endomorphisms of compact differentiable manifolds. Am. J. Math. 91, 175–199 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shub M., Sullivan D.: Expanding endomorphisms of the circle revisited. Ergod. Theory Dynam. Syst. 5(2), 285–289 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  32. Van-Hove L.: Quelques propétés générales de l’intégral de configuration d’un systém de particles avec interaction. Physica 15, 951–961 (1949)

    Article  ADS  MATH  Google Scholar 

  33. Viana, M., Oliveira, K.: Foundations of ergodic theory, volume 151 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016)

  34. Young L.-S.: Large deviations in dynamical systems. Trans. Am. Math. Soc. 318(2), 525–543 (1990)

    MathSciNet  MATH  Google Scholar 

  35. Young, L.-S.: Ergodic theory of differentiable dynamical systems. In Real and complex dynamical systems (Hillerød, 1993), volume 464 of NATO Advanced Study Institute. Series C Mathematics and Physical Sciences, pp. 293–336. Kluwer Academic Publisher, Dordrecht (1995).

  36. Zdunik A.: Parabolic orbifolds and the dimension of the maximal measure for rational maps. Invent. Math. 99(3), 627–649 (1990)

    Article  ADS  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Pavel Bleher for suggesting this problem to us and for his several helpful comments. Many other people have also provided helpful comments and advice, including Joseph Cima, Vaughn Climenhaga, Oleg Ivrii, Benjamin Jaye, Victor Kleptsyn, Michał Misiurewicz, Boris Mityagin, Juan Rivera-Letelier, William Ross, and Maxim Yattselev. We thank the referees for their very careful reading of our paper and their helpful comments. Theorem A and its proof were the main content of Anthony Ji and Caleb He’s entry in the 2016 Siemens Competition in Math, Science, and Technology. The work of the first and fourth authors was supported by NSF Grant DMS-1348589.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, IUPUI, LD Building, Room 255, 402 North Blackford Street, Indianapolis, IN, 46202-3267, USA

    Ivan Chio

  2. Harvard University, Cambridge, MA, USA

    Caleb He

  3. Yale University, New Haven, CT, USA

    Anthony L. Ji

  4. Department of Mathematical Sciences, IUPUI, LD Building, Room 224Q, 402 North Blackford Street, Indianapolis, IN, 46202-3267, USA

    Roland K. W. Roeder

Authors
  1. Ivan Chio
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Caleb He
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anthony L. Ji
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Roland K. W. Roeder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roland K. W. Roeder.

Additional information

Communicated by J. Marklof

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chio, I., He, C., Ji, A.L. et al. Limiting Measure of Lee–Yang Zeros for the Cayley Tree. Commun. Math. Phys. 370, 925–957 (2019). https://doi.org/10.1007/s00220-019-03377-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-019-03377-9

Search

Navigation