Lipschitz embeddings of random sequences
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We develop a new multi-scale framework flexible enough to solve a number of problems involving embedding random sequences into random sequences. Grimmett et al. (Random Str Algorithm 37(1):85–99, 2010) asked whether there exists an increasing \(M\)-Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough \(M\). A closely related problem is to show that two independent Poisson processes on \(\mathbb R \) are roughly isometric (or quasi-isometric). Our approach also applies in this case answering a conjecture of Szegedy and of Peled (Ann Appl Probab 20:462–494, 2010). Our theorem also gives a new proof to Winkler’s compatible sequences problem. Our approach does not explicitly depend on the particular geometry of the problems and we believe it will be applicable to a range of multi-scale and random embedding problems.
KeywordsLipschitz embedding Rough isometry Percolation Compatible sequences
Mathematics Subject Classification60K35
We would like to thank Vladas Sidoravicius for describing his work on these problems and Peter Gács for informing us of his new results. We would also like to thank Geoffrey Grimmett and Ron Peled for introducing us to the Lipschitz embedding and rough isometry problems and to Peter Winkler for very useful discussions. We are grateful to an anonymous Referee for extremely careful reading of the manuscript and for many useful comments and suggestions.
- 1.Abért, M.: Asymptotic group theory questions. Available at http://www.math.uchicago.edu/~abert/research/asymptotic.html (2008)
- 9.Gács, P.: Clairvoyant embedding in one dimension. Preprint (2012)Google Scholar
- 10.Grimmett, G.: Three problems for the clairvoyant demon. Arxiv, preprint arXiv:0903.4749 (2009)Google Scholar
- 11.Grimmett, G.R., Holroyd, A.E.: Geometry of lipschitz percolation. Ann. Inst. H. Poincaré Probab. Statist. 48(2), 309–326 (2012)Google Scholar
- 15.Gromov, M.: Hyperbolic manifolds, groups and actions. In: Proceedings of the 1978 Stony Brook Conference, Riemann Surfaces and Related Topics, pp. 183–213 (1981)Google Scholar
- 16.Holroyd, A.E., Martin, J.: Stochastic domination and comb percolation. Arxiv, preprint arXiv: 1201.6373 (2012)Google Scholar
- 18.Kesten, H., de Lima, B., Sidoravicius, V., Vares. M.E.: On the compatibility of binary sequences. Preprint arXiv:1204.3197 (2012)Google Scholar
- 19.Kesten, H., Sidoravicius, V., Vares, M.E.: Percolation in dependent environment. Preprint (2012)Google Scholar
- 23.Sidoravicius, V.: Percolation of binary words and quasi isometries of one-dimensional random systems. Preprint (2012)Google Scholar
- 24.Winkler, P.: Dependent percolation and colliding random walks. Random Str. Algorithm 16(1), 58–84 (2000)Google Scholar