The bundle Laplacian on discrete tori

2016/07/06 by Friedli, Fabien
#Combinatorics (math.CO) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.1607.01566

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 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.

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