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 $d\ge 2$ 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.

Published in at http://dx.doi.org/10.1214/10-AOP615 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (3)

• Phase Transitions on Nonamenable Graphs
• On rough isometries of Poisson processes on the line
• Geometry of Lipschitz percolation

Cited by in corpus (3)

• Geometry of Lipschitz percolation
• Lipschitz embeddings of random sequences
• Stochastic Domination and Comb Percolation