The Annals of Applied Probability

On rough isometries of Poisson processes on the line

Ron Peled

Full-text: Open access


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

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

First available in Project Euclid: 9 March 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

Export citation


  • [1] Abért, M. (2008). Private communication. Available at
  • [2] Alon, N. and Spencer, J. H. (2000). The Probabilistic Method, 2nd ed. Wiley, New York.
  • [3] Angel, O. and Benjamini, I. (2007). A phase transition for the metric distortion of percolation on the hypercube. Combinatorica 27 645–658.
  • [4] Benjamini, I. (2005). Private communication.
  • [5] Delmotte, T. (1999). Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 181–232.
  • [6] Gromov, M. (1981). Hyperbolic manifolds, groups and actions. In Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978). Annals of Mathematics Studies 97 183–213. Princeton Univ. Press, Princeton, NJ.
  • [7] Kanai, M. (1985). Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan 37 391–413.
  • [8] Kozma, G. (2007). The scaling limit of loop-erased random walk in three dimensions. Acta Math. 199 29–152.
  • [9] Kozma, G. (2006). Private communication.