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

Selected pseudo-random-number generators are applied to a Monte Carlo study of the two-dimensional square-lattice site percolation model. A generator suitable for high precision calculations is identified from an application specific test of randomness. After extended computation and analysis, an ostensibly reliable value of $p_c=0.59274598\left(4\right)$ is obtained for the percolation threshold.

Figure 1

Standard (left-hand side) and (example) nonstandard (right-hand side) enumerations of the $L=4$ square lattice.

Figure 2

Parametrized fit of first-order scaling theory [Eq. (7) with $b=0$] to experimental data [$p_m\left(L\right)$ of Table ] for the median-$p$ critical point estimator.

Figure 3

Parametrized fit of first-order scaling theory [Eq. (8) with $c=0$] to experimental data [$p_{\mathrm{cc}}\left(L\right)$ of Table ] for the cell-to-cell critical point estimator.

Figure 4

Parametrized fit of scaling theory [Eq. (9)] to experimental data [$p_h\left(L\right)$ of Table ] for the linear combination critical point estimator.

Figure 5

Parametrized fit of scaling theory [Eq. (10)] to experimental data [$p_r\left(L\right)$ of Table ] for the renormalization group fixed point percolation threshold estimator.

Figure 6

Parametrized fit of empirical power-law model $p_m\left(L\right)=p_m^{*}-a{L}^z$ to combined experimental data from the Mersenne twister, subtract with borrow, generalized xorshift, and both quad-tap generators.