Rigorous confidence intervals for critical probabilities

Abstract

We use the method of Balister, Bollobás and Walters [3] 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.

Critical Surfaces for General Bond Percolation Problems

“…This estimate agrees much better with the numerical result and falls within the confidence interval of Ref. [12]. In general, however, it becomes increasingly difficult to constrain systems with greater numbers of distinct probabilities, since the set of coefficients quickly becomes large.…”

“…Although Wu's [11]. c Reference [12]. d 1 ÿ 3p 2 ÿ 6p 3 12p 4 ÿ 6p 5 p 6 0. e 1 ÿ 3p 4 ÿ 6p 5 3p 6 15p 7 ÿ 15p 8 4p 9 0. f Equation (8).…”

Pseudo-random-number generators and the square site percolation threshold

“…The MT19937 algorithm has been used for computing integrals in semi-rigorous work by Balister, Bollobás, Walters and Riordan [35,36], and for Monte Carlo sampling by Lee [37].…”

Percolation Theory

“…Riordan and Walters [15] developed a method for computing very narrow confidence interval estimates with extremely high confidence levels, such as error probabilities of one-millionth. The confidence intervals are based on simulations, but do not use any extrapolation.…”