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      On rough isometries of Poisson processes on the line

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          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.

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          Parabolic Harnack inequality and estimates of Markov chains on graphs

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            The scaling limit of loop-erased random walk in three dimensions

            Gady Kozma (2007)
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              A phase transition for the metric distortion of percolation on the hypercube

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                Author and article information

                Journal
                14 September 2007
                2010-10-06
                Article
                10.1214/09-AAP624
                0709.2383
                ede12cb7-5b4a-4967-b2a9-f60b6b37a1f9

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                IMS-AAP-AAP624
                Annals of Applied Probability 2010, Vol. 20, No. 2, 462-494
                Published in at http://dx.doi.org/10.1214/09-AAP624 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
                math.PR
                vtex

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