Zeta functions, heat kernels and spectral invariants under
degeneration of discrete tori (pdf)
(with G. Chinta and J. Jorgenson) To appear in Nagoya Math. J.
By a discrete torus we mean the Cayley graph associated to a finite product of finite cycle
groups with generating set given by choosing a generator for each cyclic factor. In this article
we study 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
re-scaling, 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.
Applications of heat kernels on abelian groups:
ζ(2n), quadratic reciprocity, Bessel integrals
To appear in The Serge Lang Memorial Volume, Springer Verlag.
The discussion centers around three applications of heat kernel considerations
on R, Z and their quotients. These are Eulerís formula for zeta(2n), Gaussí
quadratic reciprocity law, and the evaluation of certain integrals of Bessel functions.
Some further applications are mentioned, including the functional equation of Riemannís
zeta-function, the reflection formula for the Gamma-function, and certain infinite
sums of Bessel functions.