Asymptotic Analysis Of Determinant Of Discrete Laplacian

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 precis...

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Bibliographic Details
Published in:arXiv.org 2019-10
Main Authors: Hou, Yuhang, Kandel, Santosh
Format: Article
Language:English
Subjects:
Asymptotic series
Boundary conditions
Dirichlet problem
Field theory
Hypercubes
Lattices
Partitions
Partitions (mathematics)
Toruses
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1910.02887