Journal of Statistical Physics

, Volume 122, Issue 2, pp 279–302 | Cite as

Improved Lower Bounds for the Critical Probability of Oriented Bond Percolation in Two Dimensions



We present a coupled decreasing sequence of random walks on Z that dominate the edge process of oriented bond percolation in two dimensions. Using the concept of random walk in a strip, we describe an algorithm that generates an increasing sequence of lower bounds that converges to the critical probability of oriented percolation pc. From the 7th term on, these lower bounds improve upon 0.6298, the best rigorous lower bound at present, establishing 0.63328 as a rigorous lower bound for pc. Finally, a Monte Carlo simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for pc.

Key Words

Oriented percolation Discrete time contact processes Critical probability Edge process Markov chain in a strip Coupling Simulation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Durrett, Oriented Percolation in Two Dimensions. The Annals of Probability 12(4):999–1040 (1984).MATHMathSciNetGoogle Scholar
  2. 2.
    R. Durrett, Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole (1988).Google Scholar
  3. 3.
    G. Grimmett, Percolation. Springer Verlag (1999).Google Scholar
  4. 4.
    G. Fayolle, V.A. Malyshev, M.V. Menshikov, Topics in the Constructive Theory of Countable Markov Chains. Cambridge University Press (1995).Google Scholar
  5. 5.
    R. G. Bartle, The Elements of Integration and Lebesgue Measure. John Willey & Sons Inc (1995).Google Scholar
  6. 6.
    J. H. Wilkinson, Rounding Errors in Algebraic Processes. Her Majesty's Stationary Office (1963).Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Universidade de São Paulo Instituto de Matemática e EstatísticaSão Paulo SPBrazil

Personalised recommendations