Lattice embeddings in percolation

Abstract

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.