Skip to main content
SpringerLink
Log in

Heat kernels on regular graphs and generalized Ihara zeta function formulas

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We establish a new formula for the heat kernel on regular trees in terms of classical \(I\)-Bessel functions. Although the formula is explicit, and a proof is given through direct computation, we also provide a conceptual viewpoint using the horocyclic transform on regular trees. From periodization, we then obtain a heat kernel expression on any regular graph. From spectral theory, one has another expression for the heat kernel as an integral transform of the spectral measure. By equating these two formulas and taking a certain integral transform, we obtain as application several generalized versions of the determinant formula for the Ihara zeta function associated to finite or infinite regular graphs. Our approach to the Ihara zeta function and determinant formula through heat kernel analysis follows a similar methodology which exists for quotients of rank one symmetric spaces.

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. Ahumada, G.: Fonctions périodiques et formule des traces de Selberg sur les arbres. C. R. Acad. Sci. Paris Sér. I Math 305(16), 709–712 (1987)

    MATH  MathSciNet  Google Scholar 

  2. Bass, H.: The Ihara-Selberg zeta function of a tree lattice. Int. J. Math. 3(6), 717–797 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bednarchak, D.: Heat kernel for regular trees. In harmonic functions on trees and buildings (New York, 1995), volume 206 of Contemp. Math. pp. 111–112. Am. Math. Soc. Providence, RI (1997)

  4. Chinta, G., Jorgenson, J., Karlsson, A.: Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori. Nagoya Math. J. 198, 121–172 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cowling, M., Meda, S., Setti, A.G.: Estimates for functions of the Laplace operator on homogeneous trees. Trans. Am. Math. Soc. 352(9), 4271–4293 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Clair, B., Mokhtari-Sharghi, S.: Zeta functions of discrete groups acting on trees. J. Algebra 237(2), 591–620 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chung, F., Yau, S.T.: Coverings, heat kernels and spanning trees. Electron. J. Comb. 6: Research Paper vol. 12, p. 21 (electronic) (1999)

  8. Deitmar, A.: Bass-Ihara zeta functions for non-uniform tree lattices. arXiv:1402.4759 [math.NT] (2014)

  9. Deitmar, A.: Ihara zeta functions of infinite weighted graphs, arXiv:1402.4945 [math.NT] (2014)

  10. Dodziuk, J., Mathai, V.: Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians. In the ubiquitous heat kernel, volume 398 of Contemp. Math. pp. 69–81. Am. Math. Soc. Providence, RI (2006)

  11. Dodziuk, J.: Elliptic operators on infinite graphs. Analysis. geometry and topology of elliptic operators. World Sci. Publ, NJ (2006)

    Google Scholar 

  12. Foata, D., Zeilberger, D.: A combinatorial proof of Bass’s evaluations of the Ihara-Selberg zeta function for graphs. Trans. Am. Math. Soc. 351(6), 2257–2274 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Guido, D., Isola, T., Lapidus, M.L.: Ihara’s zeta function for periodic graphs and its approximation in the amenable case. J. Funct. Anal. 255(6), 1339–1361 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  14. Grigorchuk, R.I., Żuk, A.: The Ihara zeta function of infinite graphs, the KNS spectral measure and integrable maps. In Random walks and geometry. Walter de Gruyter GmbH and Co. KG, Berlin (2004)

    Google Scholar 

  15. Hashimoto, K.: Zeta functions of finite graphs and representations of \(p\)-adic groups. In Automorphic forms and geometry of arithmetic varieties, volume 15 of Adv. Stud. Pure Math. pp. 211–280. Academic Press, Boston (1989)

  16. Horton, M.D., Newland, D.B., Terras, A.A.: The contest between the kernels in the Selberg trace formula for the \((q+1)\)-regular tree. In the ubiquitous heat kernel, volume 398 of Contemp. Math. pp. 265–293. Am. Math. Soc. Providence, RI (2006)

  17. Ihara, Y.: On discrete subgroups of the two by two projective linear group over \({\mathfrak{p}}\)-adic fields. J. Math. Soc. Japan 18, 219–235 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jorgenson, J., Lang, S.: The ubiquitous heat kernel. In mathematics unlimited-2001 and beyond, pp. 655–683. Springer, Berlin (2001)

  19. Karlsson, A., Neuhauser, M.: Heat kernels, theta identities, and zeta functions on cyclic groups. In topological and asymptotic aspects of group theory, volume 394 of Contemp. Math. pp. 177–189. Am. Math. Soc. Providence, RI (2006)

  20. Kotani, M., Sunada, T.: Zeta functions of finite graphs. J. Math. Sci. Univ. Tokyo 7(1), 7–25 (2000)

    MATH  MathSciNet  Google Scholar 

  21. McKean, H.P.: Selberg’s trace formula as applied to a compact Riemann surface. Comm. Pure Appl. Math. 25, 225–246 (1972)

    Article  MathSciNet  Google Scholar 

  22. Mnëv, P.: Discrete path integral approach to the Selberg trace formula for regular graphs. Commun. Math. Phys. 274(1), 233–241 (2007)

    Article  MATH  Google Scholar 

  23. Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bull. Lond. Math. Soc. 21(3), 209–234 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  24. Oberhettinger, F., Badii, L.: Tables of Laplace transforms. Springer, New York (1973)

    Book  MATH  Google Scholar 

  25. Oren, I., Godel, A., Smilansky, U.: Trace formulae and spectral statistics for discrete Laplacians on regular graphs (I) 4. J. Phys. A Math. Theor. 42(20), 415101 (2009)

    Article  MathSciNet  Google Scholar 

  26. Pal’tsev, B.V.: On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65(5), 681–692 (1999)

    Article  MathSciNet  Google Scholar 

  27. Serre, J.-P.: Répartition asymptotique des valeurs propres de l’opérateur de Hecke \(T_p\). J. Am. Math. Soc. 10(1), 75–102 (1997)

    Article  MATH  Google Scholar 

  28. Serre, J.-P.: Trees. Springer monographs in mathematics. Springer, Berlin. Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation (2003)

  29. Sunada, T.: \(L\)-functions in geometry and some applications. In curvature and topology of Riemannian manifolds (Katata, 1985), volume 1201 of Lecture Notes in Math. pp. 266–284. Springer, Berlin (1986)

  30. Sunada, T.: Fundamental groups and Laplacians [MR0922018 (89d:58128)]. In selected papers on number theory, algebraic geometry, and differential geometry, volume 160 of Am. Math. Soc. Trans. Ser. vol. 2, pp. 19–32. Am. Math. Soc. Providence, RI (1994)

  31. Sunada, T.: Discrete geometric analysis. In analysis on graphs and its applications, volume 77 of Proc. Sympos. Pure Math. pp. 51–83. Am. Math. Soc. Providence, RI (2008)

  32. Terras, A.: Zeta functions of graphs, volume 128 of Cambridge Studies in advanced mathematics. Cambridge University Press, Cambridge (2011). A stroll through the garden

    Google Scholar 

  33. Terras, A.A., Stark, H.M.: Zeta functions of finite graphs and coverings III. Adv. Math. 208(1), 467–489 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  34. Terras, A., Wallace, D.: Selberg’s trace formula on the \(k\)-regular tree and applications. Int. J. Math. Math. Sci. 8, 501–526 (2003)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank Anton Deitmar, Fabien Friedli and Pierre de la Harpe for reading parts of this paper and alerting us about some inaccuracies.

Author information

Authors and Affiliations

  1. Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, NY, 10031, USA

    G. Chinta & J. Jorgenson

  2. Mathematics Department, University of Geneva, 1211, Geneva, Switzerland

    A. Karlsson

Authors
  1. G. Chinta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. Jorgenson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Karlsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Karlsson.

Additional information

Communicated by A. Constantin.

G. Chinta and J. Jorgenson acknowledge support provided by Grants from the National Science Foundation and the Professional Staff Congress of the City University of New York. A. Karlsson received support from SNSF grant 200021_132528/1. Support from Institut Mittag-Leffler (Djursholm, Sweden) is also gratefully acknowledged by all authors.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chinta, G., Jorgenson, J. & Karlsson, A. Heat kernels on regular graphs and generalized Ihara zeta function formulas. Monatsh Math 178, 171–190 (2015). https://doi.org/10.1007/s00605-014-0685-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-014-0685-4

Keywords

Mathematics Subject Classification

Search

Navigation