Statistics of the percolation threshold

The statistical distribution of the percolation threshold in two-dimensional samples of finite size is investigated by a Monte Carlo method. Conducting elements are added at random positions in an N*N array, the percolation threshold being identified as the fractional area occupied by the conductor when the first continuous conducting path can be traced across the array. Studies were made on arrays having eightfold coordination with N=5, 10, 20, 40 and on arrays having fourfold coordination with N=20. The average value of the percolation threshold is about 0.4 for eightfold coordination and 0.6 for fourfold coordination. For eightfold coordination the width of the distribution decreases with array size as N-1v(p)/ with vp=1.78+or-0.03. This is a slower rate of decrease than for arrays with fourfold coordination for which various authors report vp=1.33-1.36. The difference is surprising since it is not consistent with the assumption of universality. Extrapolation of the data suggests that the percolation threshold for arrays with eightfold coordination can be determined experimentally to a precision of 1% by taking N approximately 2000.

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