Random Lattices and Random Sphere Packings: Typical Properties

TL;DR

This paper reviews properties of typical lattices and random sphere packings in high-dimensional spaces, showing their densities decrease exponentially with dimension and providing estimates for packing densities after modifications.

Contribution

It introduces new estimates for the density of random sphere packings derived from typical lattice properties in high dimensions.

Findings

Density of typical lattices is about 2^{-n}.

Modified random packings have density C(ν) 2^{-n}.

Provides bounds on the constant C(ν).

Abstract

We review results about the density of typical lattices in $R^n.$ They state that such density is of the order of $2^{-n}.$ We then obtain similar results for random packings in $R^n$: after taking suitably a fraction $\nu$ of a typical random packing $\sigma$, the resulting packing $\tau$ has density $C(\nu) 2^{-n},$ with a reasonable $C(\nu).$ We obtain estimates on $C(\nu).$