Skip to main content Accessibility help
×
Home

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

  • Béla Bollabás (a1) and Alan Stacey (a1)

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.

Copyright

Corresponding author

Postal address: Department of Pure Mathematics and Mathematical Statistics, Cambridge University, 16 Mill Lane, Cambridge, UK.

References

Hide All
[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, 201220.
[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, 573589.
[3] Broadbent, S. R. and Hammersley, J. M. (1957) Percolation processes I, II. Proc. Camb. Phil. Soc. 53, 629641, 642-645.
[4] Dhar, D. (1982) Percolation in two and three dimensions I. J. Phys. A 15, 18491858.
[5] Dhar, D. and Barma, M. (1981) Monte Carlo simulation of directed percolation on a square lattice. J. Phys. C 14, L1L6.
[6] Durrett, R. (1984) Oriented percolation in two dimensions. Ann. Prob. 12, 9991040.
[7] Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth and Brooks/Cole, New York.
[8] Durrett, R. (1992) Stochastic growth models: bounds on critical values. J. Appl. Prob. 29, 1120.
[9] Grimmett, G. (1989) Percolation. Springer, New York.
[10] Harris, T. E. (1960) A lower bound for the critical probability in a certain percolation process. Proc. Camb. Phil. Soc. 56, 1320.
[11] Kesten, H. (1980) The critical probability of bond percolation on the square lattice equals. Commun. Math. Phys. 74, 4159.
[12] Liggett, T. M. (1995) Survival of discrete time growth models, with applications to oriented percolation. Ann. Appl. Prob. 5, 613636.
[13] Russo, L. (1978) A note on percolation. Z. Wahrscheinlichkeitsth. 43, 3948.
[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. 227245.
[15] Stacey, A. M. (1994) Bounds on the critical probability in oriented percolation models. PhD thesis. University of Cambridge.

Keywords

MSC classification

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

  • Béla Bollabás (a1) and Alan Stacey (a1)

Metrics

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