Complexity and heights of tori

TL;DR

This paper derives asymptotic formulas for the number of spanning trees in discrete tori, linking graph complexity to geometric properties of associated real tori and sphere packings.

Contribution

It provides a detailed asymptotic analysis of graph complexity for discrete tori, connecting it to the height of real tori and sphere packing conjectures.

Findings

Asymptotic formulas for spanning tree counts in discrete tori

Connection between graph complexity and real torus heights

Implications for sphere packing conjectures

Abstract

We prove detailed asymptotics for the number of spanning trees, called complexity, for a general class of discrete tori as the parameters tend to infinity. The proof uses in particular certain ideas and techniques from an earlier paper. Our asymptotic formula provides a link between the complexity of these graphs and the height of associated real tori, and allows us to deduce some corollaries on the complexity thanks to certain results from analytic number theory. In this way we obtain a conjectural relationship between complexity and regular sphere packings.