## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 20, Number 2 (2010), 462-494.

### On rough isometries of Poisson processes on the line

#### 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. Abért, 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 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 20, Number 2 (2010), 462-494.

**Dates**

First available in Project Euclid: 9 March 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1268143430

**Digital Object Identifier**

doi:10.1214/09-AAP624

**Mathematical Reviews number (MathSciNet)**

MR2650039

**Zentralblatt MATH identifier**

1205.60030

**Subjects**

Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 82B43: Percolation [See also 60K35]

**Keywords**

Rough isometry quasi-isometry Poisson process percolation matching geometry of random sets one dimension

#### Citation

Peled, Ron. On rough isometries of Poisson processes on the line. Ann. Appl. Probab. 20 (2010), no. 2, 462--494. doi:10.1214/09-AAP624. https://projecteuclid.org/euclid.aoap/1268143430