On The Distribution Of Angles Between Increasingly Many Short Lattice Vectors

TL;DR

This paper studies the distribution of angles and sphere volumes related to the shortest vectors in high-dimensional random lattices, revealing a joint Poissonian and Gaussian behavior as dimension grows.

Contribution

It introduces a novel analysis of the asymptotic distribution of angles and volumes in high-dimensional lattices, extending previous work to a broader regime.

Findings

Angles and volumes exhibit joint Poissonian and Gaussian distributions as dimension increases.

Expected values of functions evaluated at these variables converge under specified growth conditions.

Results extend understanding of geometric properties of random lattices in high dimensions.

Abstract

Following S\"odergren, we consider a collection of random variables on the space $X_n$ of unimodular lattices in dimension $n$: Normalizations of the angles between the $N = N(n)$ shortest vectors in a random unimodular lattice, and the volumes of spheres with radii equal to the lengths of these vectors. We investigate the expected values of certain functions evaluated at these random variables in the regime where $N$ tends to infinity with $n$ at the rate $N = o \left( n^{1/6} \right)$. Our main result is that as $n \longrightarrow \infty$, these random variables exhibit a joint Poissonian and Gaussian behaviour.