Rigorous confidence intervals for critical probabilities

Abstract

We use the method of Balister, Bollobas and Walters to give rigorous 99.9999% confidence intervals for the critical probabilities for site and bond percolation on the 11 Archimedean lattices. In our computer calculations, the emphasis is on simplicity and ease of verification, rather than obtaining the best possible results. Nevertheless, we obtain intervals of width at most 0.0005 in all cases.

References in corpus (9)

• A fast Monte Carlo algorithm for site or bond percolation
• Efficient Monte Carlo algorithm and high-precision results for percolation
• Critical Percolation in Finite Geometries
• Values of Brownian intersection exponents III: Two-sided exponents
• Convergence of threshold estimates for two-dimensional percolation
• Determination of the bond percolation threshold for the Kagome lattice
• Estimation of Bond Percolation Thresholds on the Archimedean Lattices
• A short proof of the Harris-Kesten Theorem
• Predictions of bond percolation thresholds for the kagomé and Archimedean $(3,12^2)$ lattices

Cited by in corpus (15)

• Pseudorandom Number Generators and the Square Site Percolation Threshold
• Estimation of Bond Percolation Thresholds on the Archimedean Lattices
• Critical surfaces for general bond percolation problems
• Universal condition for critical percolation thresholds of kagome-like lattices
• Critical surfaces for general inhomogeneous bond percolation problems
• Bond percolation thresholds on Archimedean lattices from critical polynomial roots
• Percolation in Networks with Voids and Bottlenecks
• Polynomial sequences for bond percolation critical thresholds
• Retention capacity of correlated surfaces
• Rigorous Confidence Intervals on Critical Thresholds in 3 Dimensions
• The computation of generalized percolation critical polynomials by the deletion-contraction algorithm
• Improved bounds for 1-independent percolation on ${\mathbb{Z}}^n$
• Some Exact Results on Bond Percolation
• $1$-independent percolation on ${\mathbb{Z}}^2\times K_n$
• Errata and Addenda to Mathematical Constants