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

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. $p_c\ge 0.63328$. Finally a computer simulation technique is presented; the use thereof establishes 0.64450 as a non-rigorous five-digit-precision (lower) estimate for $p_c$.

16 pages, 5 figures