Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-19T16:17:53.326Z Has data issue: false hasContentIssue false
  • English
  • Français

Zeta functions, heat kernels, and spectral asymptotics on degenerating families of discrete tori

Published online by Cambridge University Press:  11 January 2016

Gautam Chinta ,
Jay Jorgenson  and
Anders Karlsson
Gautam Chinta
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031, USAchinta@sci.ccny.cuny.edu
Jay Jorgenson
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031, USAjjorgenson@mindspring.com
Anders Karlsson
Affiliation:
Mathematics Department, Royal Institute of Technology, 100 44 Stockholm, Swedenakarl@kth.se
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle groups with the generating set given by choosing a generator for each cyclic factor. In this article we examine the spectral theory of the combinatorial Laplacian for sequences of discrete tori when the orders of the cyclic factors tend to infinity at comparable rates. First, we show that the sequence of heat kernels corresponding to the degenerating family converges, after rescaling, to the heat kernel on an associated real torus. We then establish an asymptotic expansion, in the degeneration parameter, of the determinant of the combinatorial Laplacian. The zeta-regularized determinant of the Laplacian of the limiting real torus appears as the constant term in this expansion. On the other hand, using a classical theorem by Kirchhoff, the determinant of the combinatorial Laplacian of a finite graph divided by the number of vertices equals the number of spanning trees, called the complexity, of the graph. As a result, we establish a precise connection between the complexity of the Cayley graphs of finite abelian groups and heights of real tori. It is also known that spectral determinants on discrete tori can be expressed using trigonometric functions and that spectral determinants on real tori can be expressed using modular forms on general linear groups. Another interpretation of our analysis is thus to establish a link between limiting values of certain products of trigonometric functions and modular forms. The heat kernel analysis which we employ uses a careful study of I-Bessel functions. Our methods extend to prove the asymptotic behavior of other spectral invariants through degeneration, such as special values of spectral zeta functions and Epstein-Hurwitz–type zeta functions.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 198 , June 2010 , pp. 121 - 172
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[1] Athreya, K., Modified Bessel function asymptotics via probability, Statist. Probab. Lett. 5 (1987), 325–327.Google Scholar
[2] Burton, R. and Pemantle, R., Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Prob. 21 (1993), 1329–1371.CrossRefGoogle Scholar
[3] Casartelli, M., Dall’Asta, L., Vezzani, A., and Vivo, P., Dynamical invariants in the deterministic fixed-energy sandpile, Eur. Phys. J. B 52 (2006), 91–105.Google Scholar
[4] Chang, S.-C. and Shrock, R., Some exact results for spanning trees on lattices, J. Phys. A 39 (2006), 5653–5658.Google Scholar
[5] Chiu, P., Height of flattori, Proc. Amer. Math. Soc. 125 (1997), 723–730.CrossRefGoogle Scholar
[6] Dodziuk, J., “Elliptic operators on infinite graphs,” in Krzysztof Wojciechowski 50 Years—Analysis and Geometry of Boundary Value Problems, World Scientific, Hack-ensack, NJ, 2006, 353–368.Google Scholar
[7] Dodziuk, J. and Mathai, V., “Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians,” in The Ubiquitous Heat Kernel, Contemp. Math. 398, Amer. Math. Soc, Providence, 2006, 269–297.Google Scholar
[8] Duke, W. and Imamoglu, O., Special values of multiple gamma functions, J. Théor. Nombres Bordeaux 18 (2006), 113–123.Google Scholar
[9] Duplantier, B. and David, F., Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice, J. Stat. Phys. 51 (1988), 327–434.Google Scholar
[10] Epstein, P., Zur Theorie allgemeiner Zetafunctionen, Math. Ann. 56 (1903), 615–644.CrossRefGoogle Scholar
[11] Felker, J. and Lyons, R., High-precision entropy values for spanning trees in lattices, J. Phys. A 36 (2003), 8361–8365.CrossRefGoogle Scholar
[12] Godsil, C. and Royce, G., Algebraic Graph Theory, Grad. Texts Math. 207, Springer, Berlin, 2001.CrossRefGoogle Scholar
[13] Jorgenson, J. and Lang, S., “Complex analytic properties of regularized products and series,” in Basic Analysis of Regularized Products and Series, Lecture Notes in Math. 1564, Springer, Berlin, 1993.CrossRefGoogle Scholar
[14] Jorgenson, J. and Lang, S., “The ubiquitous heat kernel,” in The Mathematics Unlimited— 2001 and Beyond, Springer, Berlin, 2001, 655–683.CrossRefGoogle Scholar
[15] Karlsson, A. and Neuhauser, M., “Heat kernels, theta identities, and zeta functions on cyclic groups,” in Topological and Asymptotic Aspects of Group Theory, Contemp. Math. 394, Amer. Math. Soc, Providence, 2006, 177–189.Google Scholar
[16] Kasteleyn, P. W., The statistics of dimers on a lattice, I, The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209–1225.CrossRefGoogle Scholar
[17] Kenyon, R., The asymptotic determinant of the discrete Laplacian, Acta Math. 185 (2000), 239–286.Google Scholar
[18] Kirchhoff, G., Uber die Auflösung der Gleichungen, auf welche man bei der Unter-suchung der linearen Verteilung galvanischer Strome gefurt wird, Ann. Phys. Chem. 72 (1847), 497–508.CrossRefGoogle Scholar
[19] Osgood, B., Phillips, R., and Sarnak, P., Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.CrossRefGoogle Scholar
[20] O’Sullivan, C., Formulas for non-holomorphic Eisenstein series, in preparation.Google Scholar
[21] Pal’tsev, B. V., Two-sided bounds uniform in the real argument and the index for modified Bessel functions, Math. Notes 65 (1999), 571–581.CrossRefGoogle Scholar
[22] Sarnak, P., “Determinants of Laplacians; heights and finiteness,” in Analysis, et cetera, Academic Press, Boston, 1990, 601–622.Google Scholar
[23] Sarnak, P. and Strombergsson, A., Minima of Epstein’s zeta function and heights of flattori, Invent. Math. 165 (2006), 115–151.CrossRefGoogle Scholar
[24] Shrock, R. and Wu, F. Y., Spanning trees on graphs and lattices in d dimensions, J. Phys. A 33 (2000), 3881–3902.CrossRefGoogle Scholar
[25] Sokal, A. and Starinets, A., Pathologies of the large-N limit for RP — 1, CPN-1, QPN-1 and mixed isovector/isotensor σ-models, Nuclear Phys. B 601 (2001), 425–502.Google Scholar