LATTICE EMBEDDINGS IN PERCOLATION
成果类型：
Article
署名作者：
Grimmett, Geoffrey R.; Holroyd, Alexander E.
署名单位：
University of Cambridge; University of British Columbia; Microsoft
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/10-AOP615
发表日期：
2012
页码：
146-161
关键词：
摘要：
Does there exist a Lipschitz injection of Z(d) into the open set of a site percolation process on Z(D), if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d = D and also when d >= 2 if the Lipschitz constant M is required to be I. Earlier work of Dirr, Dondl, Grimmett, Holroyd and Scheutzow yields a positive answer for d < D and M = 2. As a result, the above question is answered for all d, D and M. Our proof in the case d = D uses Tucker's lemma from topological combinatorics, together with the aforementioned result for d < D. One application is an affirmative answer to a question of Peled concerning embeddings of random patterns in two and more dimensions.
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