Regularized limit of determinants for discrete tori

Abstract

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 for the family of discrete graphs, jointly in the resolvent and the discretization parameter. Based on a result about interchanging regularized limits and regularized integrals, we compare the regularized limit of the log-determinants of the combinatorial Laplacian on the sequence of discrete graphs with the logarithm of the zeta determinant for the Laplace Beltrami operator on the m-dimensional torus. In a similar manner we may apply our method to compare the product of the first N non-zero eigenvalues of the Laplacian on a torus (or any other smooth manifold with an explicitly known spectrum) with the zeta-regularized determinant of the Laplacian in the regularized limit as N goes to infinity.

18 pages

References in corpus (2)

• Combinatorial Quantum Field Theory and Gluing Formula for Determinants
• Asymptotic Determinant of Discrete Laplace-Beltrami Operators

Cited by in corpus (5)

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• Spectral zeta-Functions and zeta-Regularized Functional Determinants for Regular Sturm-Liouville Operators