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We consider a combinatorial Laplace operator on a sequence of discrete graphswhich approximates the m-dimensional torus when the discretization parametertends to infinity.  We establish a polyhomogeneous expansion of the resolvent trace for thefamily of discrete graphs, jointly in the resolvent and the discretizationparameter. Based on a result about interchanging regularized limits andregularized integrals, we compare the regularized limit of the log-determinantsof the combinatorial Laplacian on the sequence of discrete graphs with thelogarithm of the zeta determinant for the Laplace Beltrami operator on them-dimensional torus.  In a similar manner we may apply our method to compare the product of thefirst N non-zero eigenvalues of the Laplacian on a torus (or any other smoothmanifold with an explicitly known spectrum) with the zeta-regularizeddeterminant of the Laplacian in the regularized limit as N goes to infinity.

Regularized limit of determinants for discrete tori