Probability Theory and Related Fields

, Volume 172, Issue 3–4, pp 1121–1179 | Cite as

Lipschitz embeddings of random fields

  • Riddhipratim BasuEmail author
  • Vladas Sidoravicius
  • Allan Sly


We consider the problem of embedding one i.i.d. collection of Bernoulli random variables indexed by \({\mathbb {Z}}^d\) into an independent copy in an injective M-Lipschitz manner. For the case \(d=1\), it was shown in Basu and Sly (Probab Theory Relat Fields 159:721–775, 2014) to be possible almost surely for sufficiently large M. In this paper we provide a multi-scale argument extending this result to higher dimensions.

Mathematics Subject Classification




This work was completed when R. B. was a graduate student at the Department of Statistics at UC Berkeley and the result in this paper appeared in Chapter 4 of his Ph.D. dissertation at UC Berkeley: Lipschitz Embeddings of Random Objects and Related Topics, 2015. R. B. gratefully acknowledges the support of UC Berkeley graduate fellowship. V. S. was supported by CNPq grant Bolsa de Produtividade. A. S. was supported by NSF grant DMS-1352013, and a Simons Investigator grant. We also thank an anonymous referee for many useful comments and suggestions that helped improve both the technical and editorial quality of the paper.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Riddhipratim Basu
    • 1
    Email author
  • Vladas Sidoravicius
    • 2
    • 3
  • Allan Sly
    • 4
  1. 1.International Centre for Theoretical SciencesTata Institute of Fundamental ResearchBangaloreIndia
  2. 2.Courant Institute of Mathematical SciencesNew YorkUSA
  3. 3.NYU-ECNU Institute of Mathematical Sciences at NYU ShanghaiShanghaiChina
  4. 4.Department of MathematicsPrinceton UniversityPrincetonUSA

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