Poisson approximation of random lattices

TL;DR

This paper demonstrates that the distribution of points in a random lattice within a subset of Euclidean space closely approximates a Poisson process, with the total variation distance decreasing exponentially as dimension increases.

Contribution

It establishes an exponential decay bound on the total variation distance between a random lattice's points and a Poisson process in high dimensions.

Findings

Total variation distance is at most C e^{-c' n}

Poisson approximation becomes accurate in high dimensions

Results hold for subsets with volume at most c n and no symmetric points

Abstract

Fix a subset $S \subset \mathbb{R}^n$ of volume at most $c n$ that satisfies $S \cap (-S) = \emptyset$. We consider two point processes in $S$: the first is the Poisson point process of intensity one, and the second is the restriction of a random lattice to $S$, where the random lattice is distributed uniformly in the space of covolume-one lattices. We show that the total variation distance between these two point processes is at most $C e^{-c' n}$, where $c, C, c' > 0$ are universal constants.