### 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 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.

Original language | English (US) |
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Pages (from-to) | 721-775 |

Number of pages | 55 |

Journal | Probability Theory and Related Fields |

Volume | 159 |

Issue number | 3-4 |

DOIs | |

State | Published - Jan 1 2014 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Analysis
- Statistics and Probability
- Statistics, Probability and Uncertainty

### Keywords

- Compatible sequences
- Lipschitz embedding
- Percolation
- Rough isometry

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## Cite this

*Probability Theory and Related Fields*,

*159*(3-4), 721-775. https://doi.org/10.1007/s00440-013-0519-7