Related papers: The bundle Laplacian on discrete tori
We consider the combinatorial Laplacian on a sequence of discrete tori which approximate the m-dimensional torus. In the special case m=1, Friedli and Karlsson derived an asymptotic expansion of the corresponding spectral zeta function in…
We consider a combinatorial Laplace operator on a sequence of discrete graphs which approximates the m-dimensional torus when the discretization parameter tends to infinity. We establish a polyhomogeneous expansion of the resolvent trace…
We establish an asymptotic relation between the spectrum of the discrete Laplacian associated to discretizations of a half-translation surface with a flat unitary vector bundle and the spectrum of the Friedrichs extension of the Laplacian…
This paper presents a comprehensive analysis of the spectral properties of the connection Laplacian for both real and discrete tori. We introduce novel methods to examine these eigenvalues by employing parallel orthonormal basis in the…
We study the asymptotic expansion of the determinant of the graph Laplacian associated to discretizations of a half-translation surface endowed with a flat unitary vector bundle. By doing so, over the discretizations, we relate the…
The spectral zeta function of the Laplacian on self-similar fractal sets has been previously studied and shown to meromorphically extend to the complex plane. In this work we establish under certain conditions a relationship between the…
In this paper, we study the relation between the partition function of the free scalar field theory on hypercubes with boundary conditions and asymptotics of discrete partition functions on a sequence of "lattices" which approximate the…
For $\Pi \subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $L\Pi \cap \mathbb{Z}^2$ with Dirichlet…
Consider a surface $\Omega$ with a boundary obtained by gluing together a finite number of equilateral triangles, or squares, along their boundaries, equipped with a flat unitary vector bundle. Let $\Omega^{\delta}$ be the discretization of…
This paper generalizes Bass' work on zeta functions for uniform tree lattices. Using the theory of von Neumann algebras, machinery is developed to define the zeta function of a discrete group of automorphisms of a bounded degree tree. The…
This paper discusses the question whether the discrete spectrum of the Laplace-Beltrami operator is infinite or finite. The borderline-behavior of the curvatures for this problem will be completely determined.
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete…
We show that the Ruelle dynamical zeta function on a closed odd dimensional locally symmetric space twisted by an arbitrary flat vector bundle has a meromorphic extension to the whole complex plane and that its leading term in the Laurent…
We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extension) of the Laplacian on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle…
We give an integral representaion of the zeta-reguralized determinant of Laplacians on three dimensional Heisenberg manifolds, and study a behaivior of the values when we deform the uniform discrete subgroups. Heiseberg manifolds are the…
In this monograph, we lay some foundations of a theory of infinite dimensional Euclidean lattices - and more generally, of infinite dimensional Hermitian vector bundles over some "arithmetic curve" ${\rm Spec}\,\mathcal{O}_K$ attached to…
In this paper, we compute asymptotics for the determinant of the combinatorial Laplacian on a sequence of $d$-dimensional orthotope square lattices as the number of vertices in each dimension grows at the same rate. It is related to the…
Given a Hermitian vector bundle over an infinite weighted graph, we define the Laplacian associated to a unitary connection on this bundle and study the essential self-adjointness of a perturbation of this Laplacian by an operator-valued…
We determine the special values at positive integers of the spectral zeta function associated with the combinatorial Laplacian on the regular tree. These values admit explicit formulas in terms of certain polynomials, which we show to be…
In this paper we obtain precise asymptotics for certain families of graphs, namely circulant graphs and degenerating discrete tori. The asymptotics contain interesting constants from number theory among which some can be interpreted as…