Influence of Boundary Conditions on the Fraction of Spanning Clusters

Abstract

We use the Hoshen–Kopelman algorithm with the Nakanashi method of recycling redundant labels to measure the fraction of spanning configurations, R(pc), at and near pc, for random site percolation in two and three dimensions with different boundary conditions. For the square and cubic lattices we find that R(pc) is 0.50 and 0.28 for free edges and 0.64 (2-d) and 0.56 (3-d) for both helical and periodic boundary conditions. The error bars are of the order of ±0.01 for these results.

Correction-to-scaling exponent for two-dimensional percolation

“…For L/ = 1 or R/R 1 = e 2π , τ = 2i and Eq. (3) gives Π(2i) ≈ 0.636 454 001, which agrees closely with the measured values 0.636 65(8) of Hovi and Aharony [25], 0.638 of Acharyya and Stauffer [26], 0.64(1) of Ford, Hunter, and Jan [27], 0.6365(1) of Shchur [28], and 0.6363(3) (average) by Pruessner and Moloney [29].…”

“…., in agreement with the precise value 0.876 657 (45) found by de Oliveira, Nóbrega, and Stauffer [30]. For L/ = 1 or R/R 1 = e 2π , τ = 2i and (3) gives Π(2i) ≈ 0.636 454 001, which agrees closely with the measured values 0.636 65(8) of Hovi and Aharony [31], 0.63 of Gropengiesser and Stauffer [32], 0.638 of Acharyya and Stauffer [33], 0.64(1) of Ford, Hunter, and Jan [34], 0.6365(1) of Shchur [35], and 0.6363(3) (average) by Pruessner and Moloney [36]. [37], as follows from Cardy's original crossing formula [38].…”

Reexamination of Seven-Dimensional Site Percolation Threshold

“…The bulk p c value is independent of the boundary conditions, and so may be the powers of L in finite-size corrections, but finite-size amplitudes as well as the fraction of samples spanning at p c depend on such details. 9 The Hoshen-Kopelman algorithm stores only one line of a square lattice at any one moment, and then it is simplest to use free boundaries also in this direction. In higher dimensions it is practical to store the L d−1 sites of the one hyperplane kept in memory by a one-dimensional index i = 1, 2, .…”