Date of this Version
Random Structures & Algorithms
Draw planes in ℝ3 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.
This is the peer reviewed version of the following article: Jonasson, J., Mossel, E. and Peres, Y. (2000), Percolation in a dependent random environment. Random Struct. Alg., 16: 333–343., which has been published in final form at doi: 10.1002/1098-2418(200007)16:4<333::AID-RSA3>3.0.CO;2-C. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving http://olabout.wiley.com/WileyCDA/Section/id-820227.html#terms.
Jonasson, J., Mossel, E., & Peres, Y. (2000). Percolation in a Dependent Random Environment. Random Structures & Algorithms, 16 (4), 333-343. http://dx.doi.org/10.1002/1098-2418(200007)16:4<333::AID-RSA3>3.0.CO;2-C
Date Posted: 27 November 2017
This document has been peer reviewed.