# zbMATH — the first resource for mathematics

Stochastic domination and comb percolation. (English) Zbl 1291.60204
Summary: 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.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B43 Percolation
Full Text: