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

Abstract

Abstract. 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.

The bundle Laplacian on discrete tori

“…In statistical physics in particular, it is often interesting to look at sequences of graphs whose number of vertices go to infinity and to relate the combinatorics of such sequences to continuous objects in the limit. If the graphs are discrete tori and we are interested in the number of spanning trees, this was carried out, in all dimensions, by Chinta, Jorgenson and Karlsson in [2]. They show in particular that the constant term in the asymptotics is the regularized determinant of the continuous torus.…”

Formulas for non-holomorphic Eisenstein series and for the Riemann zeta function at odd integers

“…In Section 5 we show that Theorem 1.2 may be used to prove and generalize a result in [CJK10] related to spectral zeta functions associated to a torus. A formula of Terras and Grosswald on values of the Riemann zeta function at odd integers is also proven and generalized using Theorem 1.2 in Section 7.…”

“…In the following sections we give some applications of Theorems 1.1 and 1.2. This section provides a direct proof of an identity in [CJK10] and we give some of its context here next. For a d-tuple of positive integers N = (n 1 , .…”

An explicit prime geodesic theorem for discrete tori and the hypergeometric functions

“…Because of the simplicity of the structure of the graph, their harmonic analysis are well studied. In particular, very recently, there are various results on the number of spanning trees, which is sometimes called a complexity, of the discrete tori and their degenerated ones by establishing the theory of the heat kernel on the graphs [CJK1,CJK2,CJK3,Lo1,Lo2].…”

“…It is achieved by calculating the additive zeta function ζ (d) M (s) associated with DT (d) M in two ways: One is done by using the Ihara zeta function and the other is by the theory of the heat kernel obtained in [CJK1]. In Section 3, we investigate a special case, that is, the normalized discrete torus (m,...,m) .…”