On rough isometries of Poisson processes on the line

TL;DR

This paper investigates whether independent Poisson processes on the line are almost surely rough isometric, providing partial progress by establishing bounds on the constants involved and exploring properties of monotone rough isometries.

Contribution

It proves that the quantitative version of the rough isometry conjecture can be approached with constants of order √log n, improving upon previous trivial bounds.

Findings

Constants of order √log n suffice in the quantitative rough isometry conjecture.

Constructed rough isometries are weakly monotone.

Discussed properties and lattice structure of monotone rough isometries.

Abstract

Intuitively, two metric spaces are rough isometric (or quasi-isometric) if their large-scale metric structure is the same, ignoring fine details. This concept has proven fundamental in the geometric study of groups. Ab\'{e}rt, and later Szegedy and Benjamini, have posed several probabilistic questions concerning this concept. In this article, we consider one of the simplest of these: are two independent Poisson point processes on the line rough isometric almost surely? Szegedy conjectured that the answer is positive. Benjamini proposed to consider a quantitative version which roughly states the following: given two independent percolations on $\mathbb {N}$, for which constants are the first $n$ points of the first percolation rough isometric to an initial segment of the second, with the first point mapping to the first point and with probability uniformly bounded from below? We prove that the original question is equivalent to proving that absolute constants are possible in this quantitative version. We then make some progress toward the conjecture by showing that constants of order $\sqrt{\log n}$ suffice in the quantitative version. This is the first result to improve upon the trivial construction which has constants of order $\log n$. Furthermore, the rough isometry we construct is (weakly) monotone and we include a discussion of monotone rough isometries, their properties and an interesting lattice structure inherent in them.