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Regularized limit of determinants for discrete tori

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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 \(N\in \mathbb {N}\) 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 \(N\rightarrow \infty \).

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Notes

  1. In contrast to [14, Lemma 3.3] we do not assume that f(z, 1) and f(1, n) are smooth at \(z,n=0\), and require partial asymptotics as \(z,n\rightarrow \infty \) instead. In fact the latter result is applied to some homogeneous f(z, n), where smoothness of f(z, 1) and f(1, n) at \(z,n=0\) indeed fails.

  2. Obtained by iterating the standard Euler Maclaurin formula in one single summation parameter.

  3. Note that \(\left( \mathrm {\omega }(1,y) + (z/n)^2\right) ^{-m}\) lifts to a polyhomogeneous function on the blowup space \([[0,1]^m\times \mathbb {R}^+, \mathscr {A}]\), blown up at the corners \(\mathscr {A}=\{(y,t) \mid y_j \in \{0,1\}, t \in \{0,\infty \}\}\). Pushforward theorem of Melrose [15, 16] then yields an asymptotic expansion of the integral as \(t\rightarrow 0\) or \(t\rightarrow \infty \). The explicit structure of the expansion is irrelevant in our discussion.

  4. Note from (1.3) that \(\ker \Delta _n\) and \(\ker \Delta \) are both one-dimensional.

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Acknowledgements

The author gratefully acknowledges helpful discussions with Matthias Lesch, Nicolai Reshetikhin and Daniel Grieser. He also gratefully acknowledges financial support by the Hausdorff Center for Mathematics in Bonn and by the Mathematical Institute at Münster University.

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  1. Institute of Mathematics, Carl von Ossietzky University of Oldenburg, Ammerländer Heerstraße 114, 26129, Oldenburg, Germany

    Boris Vertman

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Communicated by A. Constantin.

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Vertman, B. Regularized limit of determinants for discrete tori. Monatsh Math 186, 539–557 (2018). https://doi.org/10.1007/s00605-017-1083-5

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