Premium
Percolation in a dependent random environment
Author(s) -
Jonasson Johan,
Mossel Elchanan,
Peres Yuval
Publication year - 2000
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/1098-2418(200007)16:4<333::aid-rsa3>3.0.co;2-c
Subject(s) - percolation (cognitive psychology) , statistical physics , percolation theory , mathematics , psychology , physics , combinatorics , neuroscience , topology (electrical circuits)
Draw planes in ℝ 3 that are orthogonal to the x axis, and intersect the x axis at the points of a Poisson process with intensity λ; similarly, draw planes orthogonal to the y and z axes using independent Poisson processes (with the same intensity). Taken together, these planes naturally define a randomly stretched rectangular lattice. Consider bond percolation on this lattice where each edge of length is open with probability e − , and these events are independent given the edge lengths. We show that this model exhibits a phase transition: for large enough λ there is an infinite open cluster a.s., and for small λ all open clusters are finite a.s. We prove this result using the method of paths with exponential intersection tails , which is not applicable in two dimensions. The question whether the analogous process in the plane exhibits a phase transition is open. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 16, 333–343, 2000