| On rough isometries of Poisson processes on the line (2007) | |||||||||
Abstract | |||||||||
| Intuitively, two metric spaces are rough isometric (or quasi-isometric) if their metric structure is the same in the large scale, ignoring fine details. This concept has proved fundamental in the geometric study of groups. Ab\'ert and later Szegedy and Benjamini have posed several probabilistic questions on 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 the answer is positive. Benjamini proposed to consider a quantitative version which roughly says: 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 uniformly bounded below probability? We prove that the original question is equivalent to proving that absolute constants are possible in this quantitative version. We then make some progress towards 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.. Comment: 35 pages, 8 figures. Version 2 corrects abstract and introduction to reflect more accurate historical account of the problem | |||||||||
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