Stochastic domination and comb percolation

Abstract

There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of $\mathbb{Z}^{d-1}$ joined by a perpendicular copy) into the open set of site percolation on $\mathbb{Z}^d$, whenever the parameter p is close enough to 1 or the Lipschitz constant is sufficiently large. This is proved using several new results and techniques involving stochastic domination, in contexts that include a process of independent overlapping intervals on $\mathbb{Z}$, and first-passage percolation on general graphs.