The Annals of Probability

Lattice embeddings in percolation

Geoffrey R. Grimmett and Alexander E. Holroyd

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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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Ann. Probab., Volume 40, Number 1 (2012), 146-161.

First available in Project Euclid: 3 January 2012

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Lipschitz embedding lattice random pattern percolation quasi-isometry


Grimmett, Geoffrey R.; Holroyd, Alexander E. Lattice embeddings in percolation. Ann. Probab. 40 (2012), no. 1, 146--161. doi:10.1214/10-AOP615.

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