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Journal of Statistical Physics

, Volume 153, Issue 5, pp 751–762 | Cite as

On a Property of Random-Oriented Percolation in a Quadrant

  • Dmitry Zhelezov
Article
  • 120 Downloads

Abstract

Grimmett’s random-orientation percolation is formulated as follows. The square lattice is used to generate an oriented graph such that each edge is oriented rightwards (resp. upwards) with probability p and leftwards (resp. downwards) otherwise. We consider a variation of Grimmett’s model proposed by Hegarty, in which edges are oriented away from the origin with probability p, and towards it with probability 1−p, which implies rotational instead of translational symmetry. We show that both models could be considered as special cases of random-oriented percolation in the NE-quadrant, provided that the critical value for the latter is \(\frac{1}{2}\). As a corollary, we unconditionally obtain a non-trivial lower bound for the critical value of Hegarty’s random-orientation model. The second part of the paper is devoted to higher dimensions and we show that the Grimmett model percolates in any slab of height at least 3 in \(\mathbb{Z}^{3}\).

Keywords

Percolation Random orientations Phase transition 

Notes

Acknowledgements

The author is very grateful to his supervisor Professor Peter Hegarty for proposing the question, helpful discussions and for reviewing drafts over and over again. The author also deeply thanks Professor Jeff Steif for very careful proof-reading and pointing out a mistake in an earlier version. I am also grateful to an anonymous MathOverflow user j.c. for sharing the Mathematica code which eventually became the starting point for the simulations presented in Appendix.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGothenburgSweden

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