Journal of Statistical Physics

, Volume 122, Issue 2, pp 279–302

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


    • Universidade de São Paulo Instituto de Matemática e Estatística
  • Thomas Logan Ritchie
    • Universidade de São Paulo Instituto de Matemática e Estatística

DOI: 10.1007/s10955-005-8022-x

Cite this article as:
Belitsky, V. & Ritchie, T.L. J Stat Phys (2006) 122: 279. doi:10.1007/s10955-005-8022-x


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 percolationDiscrete time contact processesCritical probabilityEdge processMarkov chain in a stripCouplingSimulation

Copyright information

© Springer Science + Business Media, Inc. 2006