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.

Regularized Laplacian determinants of self-similar fractals

“…In this section, we show that it is a fractal with spectral zeta function such that it has no poles on the imaginary axis. Moreover, we establish a connection between the discrete and continuous determinants of the Laplacian bearing some resemblance to [ 11 ]. We depict the diamond fractal and its first graph approximation in (Fig.…”

“…On the other hand, in [ 11 ] a connection between the determinant of the discrete Laplacians and the regularized determinant has been made in the setting of the discrete Euclidean torus. Specifically, let denote a d -tuple of positive integers parametrized by , such that for each j , we have as .…”

“…Specifically, let denote a d -tuple of positive integers parametrized by , such that for each j , we have as . One then defines the d -dimensional discrete torus as the product space If A is the diagonal matrix with entries and , the authors of [ 11 ] established the formula where RT , A is the real torus , and is a specific special function. A variation of this result was also studied in [ 40 ].…”

On an approach for evaluating certain trigonometric character sums using the discrete time heat kernel

“…In this section we will study (10), which is the discrete time heat kernel on the Cayley graph X = C(G m , S, π S ), twisted by χ β . The heat kernel on X can be computed from the heat kernel K Y (x, y; n) on the Cayley graph Y = C(Z d , S, π S ) by using the methods which are common in the theory of automorphic forms, and is analogous to the construction of the continuous time heat kernel on the discrete torus; see [KN06], [CJK10] and [Do12]. We start by determining…”

Heat kernel and D n,r decomposition of some families of weakly semi-regular bipartite graphs

“…The heat kernel of a graph is obtained by exponentiating the Laplacian eigen-system with time [1]. The heat kernel of a (q + 1)-regular graph G is given by K G (t, x 0 , x) = e (q+1)t Now let us consider the SM family of graphs.…”