On the Poisson distribution of lengths of lattice vectors in a random lattice

TL;DR

This paper proves that in high-dimensional random lattices, the distribution of lengths of non-zero vectors converges to a Poisson process, extending previous results by Rogers and Schmidt.

Contribution

It generalizes earlier findings by demonstrating the Poisson distribution of lattice vector lengths in high dimensions for random lattices.

Findings

Lengths of lattice vectors form a Poisson process as dimension increases

Convergence to Poisson process is established in high-dimensional limit

Extends previous results by Rogers and Schmidt

Abstract

We prove that the volumes determined by the lengths of the non-zero vectors $\pm\vecx$ in a random lattice L of covolume 1 define a stochastic process that, as the dimension n tends to infinity, converges weakly to a Poisson process on the positive real line with intensity 1/2. This generalizes earlier results by Rogers and Schmidt.