Advertisement

Probability Theory and Related Fields

, Volume 159, Issue 3–4, pp 721–775 | Cite as

Lipschitz embeddings of random sequences

  • Riddhipratim Basu
  • Allan SlyEmail author
Article

Abstract

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.

Keywords

Lipschitz embedding Rough isometry Percolation  Compatible sequences 

Mathematics Subject Classification

60K35 

Notes

Acknowledgments

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.

References

  1. 1.
    Abért, M.: Asymptotic group theory questions. Available at http://www.math.uchicago.edu/~abert/research/asymptotic.html (2008)
  2. 2.
    Balister, P.N., Bollobás, B., Stacey, A.M.: Dependent percolation in two dimensions. Probab. Theory Relat. Fields 117, 495–513 (2000)CrossRefzbMATHGoogle Scholar
  3. 3.
    Benjamini, I., Kesten, H.: Percolation of arbitrary words in \(\{0,1\}^{ N}\). Ann. Probab. 23(3), 1024–1060 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Coppersmith, D., Tetali, P., Winkler, P.: Collisions among random walks on a graph. SIAM J. Discrete Math. 6, 363 (1993)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Gács, P.: Compatible sequences and a slow Winkler percolation. Combin. Probab. Comput. 13(6), 815–856 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Gács, P.: Clairvoyant scheduling of random walks. Random Str. Algorithm 39, 413–485 (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Gács, P.: Clairvoyant embedding in one dimension. Preprint (2012)Google Scholar
  10. 10.
    Grimmett, G.: Three problems for the clairvoyant demon. Arxiv, preprint arXiv:0903.4749 (2009)Google Scholar
  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)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Grimmett, G.R., Holroyd, A.E.: Lattice embeddings in percolation. Ann Probab. 40(1), 146–161 (2012)CrossRefzbMATHMathSciNetGoogle 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)CrossRefzbMATHMathSciNetGoogle 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)Google Scholar
  16. 16.
    Holroyd, A.E., Martin, J.: Stochastic domination and comb percolation. Arxiv, preprint arXiv: 1201.6373 (2012)Google Scholar
  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)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 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. 19.
    Kesten, H., Sidoravicius, V., Vares, M.E.: Percolation in dependent environment. Preprint (2012)Google Scholar
  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)CrossRefMathSciNetGoogle 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)MathSciNetGoogle Scholar
  22. 22.
    Peled, R.: On rough isometries of poisson processes on the line. Ann. Appl. Probab. 20, 462–494 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Sidoravicius, V.: Percolation of binary words and quasi isometries of one-dimensional random systems. Preprint (2012)Google Scholar
  24. 24.
    Winkler, P.: Dependent percolation and colliding random walks. Random Str. Algorithm 16(1), 58–84 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

Personalised recommendations