Journal of Statistical Physics

, Volume 122, Issue 2, pp 279-302

First online:

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

  • Vladimir BelitskyAffiliated withUniversidade de São Paulo Instituto de Matemática e Estatística Email author 
  • , Thomas Logan RitchieAffiliated withUniversidade de São Paulo Instituto de Matemática e Estatística

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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 p c. 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 p c. Finally, a Monte Carlo simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for p c.

Key Words

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