Skip to main content Accessibility help
×
Home

Forward Clusters for Degenerate Random Environments

  • MARK HOLMES (a1) and THOMAS S. SALISBURY (a2)

Abstract

We consider connectivity properties and asymptotic slopes for certain random directed graphs on ℤ2 in which the set of points $\mathcal{C}_o$ that the origin connects to is always infinite. We obtain conditions under which the complement of $\mathcal{C}_o$ has no infinite connected component. Applying these results to one of the most interesting such models leads to an improved lower bound for the critical occupation probability for oriented site percolation on the triangular lattice in two dimensions.

Copyright

References

Hide All
[1] Balister, P., Bollobás, B. and Stacey, A. (1994) Improved upper bounds for the critical probability of oriented percolation in two dimensions. Random Struct. Alg. 5 573589.
[2] Berger, N. and Deuschel, J.-D. (2012) Quenched invariance principle for random walk in balanced random environment. Probab. Theory Rel. Fields 152 207230.
[3] De'Bell, K. and Essam, J. W. (1983) Estimates of the site percolation probability exponents for some directed lattices. J. Phys. A 16 31453147.
[4] Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Probab. 12 9991040.
[5] Grimmett, G. and Hiemer, P. (2002) Directed percolation and random walk. In In and Out of Equilibrium: Mambucaba 2000, Vol. 51 of Progress in Probability, Birkhäuser, pp. 273297.
[6] Holmes, M. and Salisbury, T. S. (2014) Degenerate random environments. Random Struct. Alg. 45 111137.
[7] Holmes, M. and Salisbury, T. S. (2014) Random walks in degenerate random environments. Canadian J. Math. 66 10501077.
[8] Holmes, M. and Salisbury, T. S. (2016) Ballisticity and invariance principle for random walk in non-elliptic random environment. Preprint.
[9] Holmes, M. and Salisbury, T. S. (2016) Notes on oriented percolation. arXiv:1603.07806
[10] Hughes, B. D. (1995/1996) Random Walks and Random Environments, Vols 1, 2, Oxford University Press.
[11] Jensen, I. and Guttmann, A. J. (1996) Series expansions of the percolation probability on the directed triangular lattice. J. Phys. A 29 497517.

MSC classification

Related content

Powered by UNSILO

Forward Clusters for Degenerate Random Environments

  • MARK HOLMES (a1) and THOMAS S. SALISBURY (a2)

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.