# On a Property of Random-Oriented Percolation in a Quadrant

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## 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.

## References

- 1.Ballister, P., Bollobás, B., Stacey, A.: Improved upper bounds for the critical probability on oriented percolation. Random Struct. Algorithms
**5**, 573–589 (1994) CrossRefGoogle Scholar - 2.Grimmett, G.: Percolation, 1st edn. Springer, Berlin (1989) zbMATHGoogle Scholar
- 3.Grimmett, G.: Infinite paths in randomly oriented lattices. Random Struct. Algorithms
**3**, 257–266 (2000) MathSciNetGoogle Scholar - 4.Harris, T.E.: A lower bound for the critical probability in a certain percolation process. Math. Proc. Camb. Philos. Soc.
**56**, 13–20 (1960) ADSCrossRefzbMATHGoogle Scholar - 5.
- 6.Liggett, T.M., Schonmann, R.H., Stacey, A.M.: Domination by product measures. Ann. Probab.
**1**, 71–95 (1997) MathSciNetGoogle Scholar - 7.Linusson, S.: A note on correlations in randomly oriented graphs. arXiv:0905.2881v2
- 8.Xian-Yuan, W.: On the random-oriented percolation. Acta Math. Sci. Ser. B Engl. Ed.
**2**, 265–274 (2001) Google Scholar