Poisson asymptotics for random projections of points on a high-dimensional sphere

TL;DR

This paper investigates how projecting high-dimensional sphere points onto a random line results in a Poisson-like distribution, under certain separation conditions, revealing asymptotic behavior of such projections.

Contribution

It establishes conditions under which the distribution of projected points converges to a Poisson process, providing new insights into high-dimensional geometric probability.

Findings

Projection of well-separated points approximates a Poisson process

Asymptotic Poisson behavior occurs in high dimensions

Results apply to various high-dimensional geometric configurations

Abstract

Project a collection of points on the high-dimensional sphere onto a random direction. If most of the points are sufficiently far from one another in an appropriate sense, the projection is locally close in distribution to the Poisson point process.