Lipschitz embeddings of random sequences

• 成果类型: Article
• 署名作者: Basu, Riddhipratim; Sly, Allan
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0519-7
• 发表日期: 2014
• 页码: 721-775
• 关键词: dependent percolation; rough isometries; arbitrary words

摘要

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 -Lipschitz embedding from one i.i.d. Bernoulli sequence into an independent copy with positive probability. We give a positive answer for large enough . A closely related problem is to show that two independent Poisson processes on 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.