Abstract
Exact series expansion data of Sykes et al. are used to calculate the average numberc n and perimeters n of clusters of sizen≦20 in the site percolation problem for the triangular, square, and honeycomb lattice. At the percolation thresholdp n we find a sharply peaked distribution of perimeterss n with mean 〈s n 〉=((1−p n )/p c )n+O(n σ) and width 〈s 2 n 〉−〈S n 〉2∝n 1.6 whereσ≃1/βδ=0.39. This “perimeter” 〈s n 〉 should not be interpreted as a cluster “surface” in the usual sense. Two tests confirm the universality hypothesis with reasonable accuracy. The asymptotic decay of the cluster numbersc n withn is consistent with the postulated asymmetry aboutp c : logc n ∝−n ζ forn→∞ withζ≃1 forp<p c andζ≃1/2 forp>p c .
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Stauffer, D. Exact distribution of cluster size and perimeter for two-dimensional percolation. Z Physik B 25, 391–399 (1976). https://doi.org/10.1007/BF01315255
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DOI: https://doi.org/10.1007/BF01315255