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.
"Lattice embeddings in percolation." Ann. Probab. 40 (1) 146 - 161, January 2012. https://doi.org/10.1214/10-AOP615