Electronic Journal of Probability

Stochastic domination and comb percolation

Alexander Holroyd and James Martin

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 5, 16 pp.

Accepted: 8 January 2014
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B43: Percolation [See also 60K35]

stochastic domination percolation comb graph Lipschitz embedding first-passage percolation

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Holroyd, Alexander; Martin, James. Stochastic domination and comb percolation. Electron. J. Probab. 19 (2014), paper no. 5, 16 pp. doi:10.1214/EJP.v19-2806. https://projecteuclid.org/euclid.ejp/1465065647

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