Lattice embeddings in percolation

TL;DR

This paper investigates the possibility of Lipschitz embeddings of integer lattices into the open sites of percolation models, providing comprehensive results that settle the question for all dimensions and Lipschitz constants using topological combinatorics.

Contribution

It establishes negative results for embeddings when dimensions are equal and Lipschitz constant is 1, and extends positive results for lower dimensions, completing the understanding for all cases.

Findings

No Lipschitz embedding exists when d=D and M=1.

Positive embeddings exist for d<D and M=2, as previously shown.

The proof employs Tucker's lemma from topological combinatorics.

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.