Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-06T01:14:36.584Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximate upper bounds for the critical probability of oriented percolation in two dimensions based on rapidly mixing Markov chains

Part of: Special processes Graph theory

Published online by Cambridge University Press:  14 July 2016

Béla Bollabás  and
Alan Stacey
Béla Bollabás*
Affiliation:
Cambridge University
Alan Stacey*
Affiliation:
Cambridge University
*
∗Postal address: Department of Pure Mathematics and Mathematical Statistics, Cambridge University, 16 Mill Lane, Cambridge, UK.
∗Postal address: Department of Pure Mathematics and Mathematical Statistics, Cambridge University, 16 Mill Lane, Cambridge, UK.
Get access Rights & Permissions [Opens in a new window]

Abstract

We develop a technique for establishing statistical tests with precise confidence levels for upper bounds on the critical probability in oriented percolation. We use it to give pc < 0.647 with a 99.999967% confidence. As Monte Carlo simulations suggest that pc ≈ 0.6445, this bound is fairly tight.

Keywords

PERCOLATIONRAPID MIXINGCOUPLINGCOMPUTER ALGORITHM

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 05C80: Random graphs
Type
Research Papers
Information
Journal of Applied Probability , Volume 34 , Issue 4 , December 1997 , pp. 859 - 867
Copyright
Copyright © Applied Probability Trust 1997 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Balister, P., Bollobás, B. and Stacey, A. (1993) Upper bounds for the critical probability of oriented percolation in two dimensions. Proc. R. Soc. London A 440, 201–220.Google Scholar
[2] Balister, P., Bollobás, B. and Stacey, A. (1994) Improved upper bounds for the critical probability of oriented percolation in two dimensions. Rand. Struct. Alg. 5, 573–589.CrossRefGoogle Scholar
[3] Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes I, II. Proc. Camb. Phil. Soc. 53, 629–641, 642-645.CrossRefGoogle Scholar
[4] Dhar, D. (1982) Percolation in two and three dimensions I. J. Phys. A 15, 1849–1858.Google Scholar
[5] Dhar, D. and Barma, M. (1981) Monte Carlo simulation of directed percolation on a square lattice. J. Phys. C 14, L1–L6.Google Scholar
[6] Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Prob. 12, 999–1040.CrossRefGoogle Scholar
[7] Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks/Cole, New York.Google Scholar
[8] Durrett, R. (1992) Stochastic growth models: bounds on critical values. J. Appl. Prob. 29, 11–20.CrossRefGoogle Scholar
[9] Grimmett, G. (1989) Percolation. Springer, New York.CrossRefGoogle Scholar
[10] Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 13–20.CrossRefGoogle Scholar
[11] Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals. Commun. Math. Phys. 74, 41–59.CrossRefGoogle Scholar
[12] Liggett, T. M. (1995) Survival of discrete time growth models, with applications to oriented percolation. Ann. Appl. Prob. 5, 613–636.CrossRefGoogle Scholar
[13] Russo, L. (1978) A note on percolation. Z. Wahrscheinlichkeitsth. 43, 39–48.CrossRefGoogle Scholar
[14] Seymour, P. D. and Welsh, D. J. A. (1978) Percolation probabilities on the square lattice. In Advances in Graph Theory. ed. Bollobás, B. North-Holland, Amsterdam. pp. 227–245.CrossRefGoogle Scholar
[15] Stacey, A. M. (1994) Bounds on the critical probability in oriented percolation models. PhD thesis. University of Cambridge.Google Scholar