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Large simulations of 2D site percolation on a square lattice provide several estimates of the fractal dimension Dh of the 'hull' or outer boundary of percolation clusters close to criticality. Statistical estimates based on an ensemble of more than 106 clusters with varying sizes yield Dh=1.74+or-0.02. Geometrical measurements of the hull two-point correlation for individual clusters with more than 105 sites give Dh=1.76+or-0.01. The observed scaling behaviour is the same both slightly above and below pc.
Richard F. Voss (Fri,) studied this question.