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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
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