Regularized limit of determinants for discrete tori

TL;DR

This paper develops a method to compare the regularized limits of determinants of discrete Laplacians on approximating graphs with the continuous Laplace operator on tori, providing a bridge between discrete and continuous spectral analysis.

Contribution

It introduces a polyhomogeneous expansion of the resolvent trace for discrete graph Laplacians and establishes a framework for comparing their determinants with those of continuous Laplacians.

Findings

Established a polyhomogeneous expansion of the resolvent trace.

Compared regularized limits of discrete and continuous Laplacian determinants.

Applied method to eigenvalue products and zeta-regularized determinants.

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.