%0 Journal Article
%T Lattice embeddings in percolation
%A Geoffrey R. Grimmett
%A Alexander E. Holroyd
%J Mathematics
%D 2010
%I arXiv
%R 10.1214/10-AOP615
%X Does there exist a Lipschitz injection of $\mathbb{Z}^d$ into the open set of a site percolation process on $\mathbb{Z}^D$, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d=D and also when $d\geq2$ 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.
%U http://arxiv.org/abs/1003.3950v2