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Lipschitz embeddings of random sequences


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.

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  1. 1.

    Abért, M.: Asymptotic group theory questions. Available at (2008)

  2. 2.

    Balister, P.N., Bollobás, B., Stacey, A.M.: Dependent percolation in two dimensions. Probab. Theory Relat. Fields 117, 495–513 (2000)

    Article  MATH  Google Scholar 

  3. 3.

    Benjamini, I., Kesten, H.: Percolation of arbitrary words in \(\{0,1\}^{ N}\). Ann. Probab. 23(3), 1024–1060 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Coppersmith, D., Tetali, P., Winkler, P.: Collisions among random walks on a graph. SIAM J. Discrete Math. 6, 363 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    de Lima, B.N.B., Sanchis, R., Silva, R.W.C.: Percolation of words on \(\mathbb{Z}^d\) with long-range connections. J. Appl. Probab. 48(4), 1152–1162 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Dirr, N., Dondl, P.W., Grimmett, G.R., Holroyd, A.E., Scheutzow, M.: Lipschitz percolation. Electron. Comm. Probab. 15, 14–21 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Gács, P.: Compatible sequences and a slow Winkler percolation. Combin. Probab. Comput. 13(6), 815–856 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Gács, P.: Clairvoyant scheduling of random walks. Random Str. Algorithm 39, 413–485 (2011)

    Article  MATH  Google Scholar 

  9. 9.

    Gács, P.: Clairvoyant embedding in one dimension. Preprint (2012)

  10. 10.

    Grimmett, G.: Three problems for the clairvoyant demon. Arxiv, preprint arXiv:0903.4749 (2009)

  11. 11.

    Grimmett, G.R., Holroyd, A.E.: Geometry of lipschitz percolation. Ann. Inst. H. Poincaré Probab. Statist. 48(2), 309–326 (2012)

    Google Scholar 

  12. 12.

    Grimmett, G.R., Holroyd, A.E.: Plaquettes, spheres, and entanglement. Electr. J. Probab. 15, 1415–1428 (2010)

    MATH  MathSciNet  Google Scholar 

  13. 13.

    Grimmett, G.R., Holroyd, A.E.: Lattice embeddings in percolation. Ann Probab. 40(1), 146–161 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Grimmett, G.R., Liggett, T.M., Richthammer, T.: Percolation of arbitrary words in one dimension. Random Str. Algorithm 37(1), 85–99 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. 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)

  16. 16.

    Holroyd, A.E., Martin, J.: Stochastic domination and comb percolation. Arxiv, preprint arXiv: 1201.6373 (2012)

  17. 17.

    Kanai, M.: Rough isometries, and combinatorial approximations of geometries of non. compact riemannian manifolds. J. Math. Soc. Jpn. 37(3), 391–413 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    Kesten, H., de Lima, B., Sidoravicius, V., Vares. M.E.: On the compatibility of binary sequences. Preprint arXiv:1204.3197 (2012)

  19. 19.

    Kesten, H., Sidoravicius, V., Vares, M.E.: Percolation in dependent environment. Preprint (2012)

  20. 20.

    Kesten, H., Sidoravicius, V., Zhang, Y.: Almost all words are seen in critical site percolation on the triangular lattice. Elec. J. Probab. 3, 1–75 (1998)

    Article  MathSciNet  Google Scholar 

  21. 21.

    Kesten, H., Sidoravicius, V., Zhang, Y.: Percolation of arbitrary words on the close-packed graph of \(\mathbb{Z}^2\). Electr. J. Probab. (electronic) 6, 4–27 (2001)

    MathSciNet  Google Scholar 

  22. 22.

    Peled, R.: On rough isometries of poisson processes on the line. Ann. Appl. Probab. 20, 462–494 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  23. 23.

    Sidoravicius, V.: Percolation of binary words and quasi isometries of one-dimensional random systems. Preprint (2012)

  24. 24.

    Winkler, P.: Dependent percolation and colliding random walks. Random Str. Algorithm 16(1), 58–84 (2000)

    Google Scholar 

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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.

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Correspondence to Allan Sly.

Additional information

R. Basu was supported by Loéve Fellowship, Department of Statistics, University of California, Berkeley. A. Sly was supported by NSF grant DMS-1208338 and a Sloan Fellowship.

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Basu, R., Sly, A. Lipschitz embeddings of random sequences. Probab. Theory Relat. Fields 159, 721–775 (2014).

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  • Lipschitz embedding
  • Rough isometry
  • Percolation
  • Compatible sequences

Mathematics Subject Classification

  • 60K35