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Annales de l’Institut Henri Poincaré D


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Volume 6, Issue 1, 2019, pp. 97–121
DOI: 10.4171/AIHPD/66

Published online: 2018-08-20

The bundle Laplacian on discrete tori

Fabien Friedli[1]

(1) Université de Genève, Switzerland

We prove an asymptotic formula for the determinant of the bundle Laplacian on discrete $d$-dimensional tori as the number of vertices tends to infinity. This determinant has a combinatorial interpretation in terms of cycle-rooted spanning forests. We also establish a relation (in the limit) between the spectral zeta function of a line bundle over a discrete torus, the spectral zeta function of the infinite graph $\mathbb Z^d$ and the Epstein–Hurwitz zeta function. The latter can be viewed as the spectral zeta function of the twisted continuous torus which is the limit of the sequence of discrete tori.

Keywords: Bundle Laplacian, heat kernel, cycle-rooted spanning forest, spectral zeta function, Kronecker limit formula

Friedli Fabien: The bundle Laplacian on discrete tori. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), 97-121. doi: 10.4171/AIHPD/66