On the distribution of angles between the N shortest vectors in a random lattice

TL;DR

This paper investigates the asymptotic distribution of angles and lengths among the shortest vectors in high-dimensional random lattices, linking results to spectral properties of flat tori and minima distribution.

Contribution

It provides the joint distribution of lengths and angles of the N shortest vectors in high-dimensional lattices and interprets these results through Laplacian eigenvalues and eigenfunctions.

Findings

Distribution of angles between shortest vectors characterized as dimension grows

Connection established between lattice vectors and Laplacian spectra on flat tori

Limit distribution of successive minima analyzed in high dimensions

Abstract

We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random n-dimensional lattice as n tends to infinity. Moreover we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally we discuss the limit distribution of any finite number of successive minima of a random n-dimensional lattice as n tends to infinity.