Source: Nagoya Math. J. Volume 198
(2010), 121-172.
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.
References
[1] K. Athreya, Modified Bessel function asymptotics via probability, Statist. Probab. Lett. 5 (1987), 325–327.
Mathematical Reviews (MathSciNet):
MR903806
[2] R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impedances, Ann. Prob. 21 (1993), 1329–1371.
[3] M. Casartelli, L. Dall’Asta, A. Vezzani, and P. Vivo, Dynamical invariants in the deterministic fixed-energy sandpile, Eur. Phys. J. B 52 (2006), 91–105.
[4] S.-C. Chang and R. Shrock, Some exact results for spanning trees on lattices, J. Phys. A 39 (2006), 5653–5658.
[5] P. Chiu, Height of flat tori, Proc. Amer. Math. Soc. 125 (1997), 723–730.
[6] J. Dodziuk, “Elliptic operators on infinite graphs,” in Krzysztof Wojciechowski 50 Years—Analysis and Geometry of Boundary Value Problems, World Scientific, Hackensack, NJ, 2006, 353–368.
[7] J. Dodziuk and V. Mathai, “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.
[8] W. Duke and Ö. Imamog̈lu, Special values of multiple gamma functions, J. Théor. Nombres Bordeaux 18 (2006), 113–123.
[9] B. Duplantier and F. David, Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice, J. Stat. Phys. 51 (1988), 327–434.
Mathematical Reviews (MathSciNet):
MR952941
[10] P. Epstein, Zur Theorie allgemeiner Zetafunctionen, Math. Ann. 56 (1903), 615–644.
[11] J. Felker and R. Lyons, High-precision entropy values for spanning trees in lattices, J. Phys. A 36 (2003), 8361–8365.
[12] C. Godsil and G. Royce, Algebraic Graph Theory, Grad. Texts Math. 207, Springer, Berlin, 2001.
[13] J. Jorgenson and S. Lang, “Complex analytic properties of regularized products and series,” in Basic Analysis of Regularized Products and Series, Lecture Notes in Math. 1564, Springer, Berlin, 1993.
[14] J. Jorgenson and S. Lang, “The ubiquitous heat kernel,” in The Mathematics Unlimited—2001 and Beyond, Springer, Berlin, 2001, 655–683.
[15] A. Karlsson and M. Neuhauser, “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.
[16] P. W. Kasteleyn, The statistics of dimers on a lattice, I, The number of dimer arrangements on a quadratic lattice, Physica 27 (1961), 1209–1225.
[17] R. Kenyon, The asymptotic determinant of the discrete Laplacian, Acta Math. 185 (2000), 239–286.
[18] G. Kirchhoff, Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Strome gefurt wird, Ann. Phys. Chem. 72 (1847), 497–508.
[19] B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.
Mathematical Reviews (MathSciNet):
MR960228
[20] C. O’Sullivan, Formulas for non-holomorphic Eisenstein series, in preparation.
[21] B. V. Pal’tsev, Two-sided bounds uniform in the real argument and the index for modified Bessel functions, Math. Notes 65 (1999), 571–581.
[22] P. Sarnak, “Determinants of Laplacians; heights and finiteness,” in Analysis, et cetera, Academic Press, Boston, 1990, 601–622.
[23] P. Sarnak and A. Strombergsson, Minima of Epstein’s zeta function and heights of flat tori, Invent. Math. 165 (2006), 115–151.
[24] R. Shrock and F. Y. Wu, Spanning trees on graphs and lattices in d dimensions, J. Phys. A 33 (2000), 3881–3902.
[25] A. Sokal and A. Starinets, Pathologies of the large-N limit for RPN−1, CPN−1, QPN−1 and mixed isovector/isotensor σ-models, Nuclear Phys. B 601 (2001), 425–502.
