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

2004/12/17 by Thomas Logan Ritchie, Ritchie, Thomas Logan, Vladimir Belitsky +1
Mathematics · Physics and Astronomy · #60K35 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math-ph #math.MP #math.PR #msc:60K35

paper · pdf · doi:10.48550/arxiv.math/0412348

16 pages, 5 figures

arxiv created 2004/12/17 · openalex publication_date 2004/12/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present a coupled decreasing sequence of random walks on \mathbb Z that dominates the edge process of oriented-bond percolation in two dimensions. Using the concept of "random walk in a strip ", we construct an algorithm that generates an increasing sequence of lower bounds that converges to the critical probability of oriented-bond percolation. Numerical calculations of the first ten lower bounds thereby generated lead to an improved,i.e. higher, rigorous lower bound to this critical probability, viz. p_c ≥ 0.63328 . Finally a computer simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for p_c.

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