The bundle Laplacian on discrete tori

TL;DR

This paper derives an asymptotic formula for the determinant of the bundle Laplacian on large discrete tori, linking spectral properties with combinatorial structures and special zeta functions.

Contribution

It introduces a new asymptotic analysis of the bundle Laplacian determinant on discrete tori and connects spectral zeta functions with combinatorial and continuous models.

Findings

Asymptotic formula for the determinant as vertices grow large

Relation between spectral zeta functions of discrete and continuous tori

Connection to cycle-rooted spanning forests

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.