Asymptotic Analysis Of Determinant Of Discrete Laplacian

TL;DR

This paper investigates the asymptotic behavior of the determinant of the discrete Laplacian on hypercubes and tori, linking it to the continuous Laplacian's zeta-regularized determinant with boundary conditions.

Contribution

It establishes a precise asymptotic relation between discrete and continuous Laplacian determinants, including explicit constants, for hypercubes and tori.

Findings

Log-determinant of discrete Laplacian approaches the continuous case as mesh size decreases.

Explicit constant term in asymptotics relates to the zeta-regularized determinant.

Results extend to massive Laplacian on tori.

Abstract

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 hypercube as the mesh approaches to zero. More precisely, we show that the logarithm of the zeta regularized determinant of Laplacian on the hypercube with Dirichlet boundary condition appears as the constant term in the asymptotic expansion of the log-determinant of the discrete Laplacian up to an explicitly computable constant. We also investigate similar problems for the massive Laplacian on tori.