# The bundle Laplacian on discrete tori

### Fabien Friedli

Université de Genève, Switzerland

## Abstract

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.

## Cite this article

Fabien Friedli, The bundle Laplacian on discrete tori. Ann. Inst. Henri Poincaré Comb. Phys. Interact. 6 (2019), no. 1, pp. 97–121

DOI 10.4171/AIHPD/66