Stochastic Domination and Comb Percolation

2012/01/30 by Alexander E. Holroyd, Holroyd, Alexander E., Jean Martin +2 · 2 citations
Mathematics · #Markov Chains and Monte Carlo Methods #Random Matrices and Applications #Stochastic processes and statistical mechanics #math.PR #msc:60K35 #msc:82B43

paper · pdf · doi:10.48550/arxiv.1201.6373

21 pages

arxiv created 2012/01/30 · arxiv updated 2012/02/01

Abstract

There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^d-1 joined by a perpendicular copy) into the open set of site percolation on 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 Z, and first-passage percolation on general graphs.

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