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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, [email protected]
Jay Jorgenson
Affiliation:
Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, New York 10031, [email protected]
Anders Karlsson
Affiliation:
Mathematics Department, Royal Institute of Technology, 100 44 Stockholm, [email protected]
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Abstract

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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

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