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.
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