Open Access
January 2012 Lattice embeddings in percolation
Geoffrey R. Grimmett, Alexander E. Holroyd
Ann. Probab. 40(1): 146-161 (January 2012). DOI: 10.1214/10-AOP615


Does there exist a Lipschitz injection of ℤd into the open set of a site percolation process on ℤD, 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 1. 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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Geoffrey R. Grimmett. Alexander E. Holroyd. "Lattice embeddings in percolation." Ann. Probab. 40 (1) 146 - 161, January 2012.


Published: January 2012
First available in Project Euclid: 3 January 2012

zbMATH: 1238.60110
MathSciNet: MR2917770
Digital Object Identifier: 10.1214/10-AOP615

Primary: 60K35

Keywords: lattice , Lipschitz embedding , percolation , quasi-isometry , random pattern

Rights: Copyright © 2012 Institute of Mathematical Statistics

Vol.40 • No. 1 • January 2012
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