The authors study the form of the probability density P(r, t) of the end-to-end distance r of a segment of length t of the perimeter of the 2D percolation cluster at criticality. They find that P(r, t) behaves as P(r, t) varies as t-2dH/ exp(-(r/t1dH/)delta ), where delta =1/(1-1/dH). The exponent dH=7/4 is the fractal dimension of the hull (the external perimeter of a percolation cluster). This result fits very well with recent numerical data.
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Gouyet et al. (1988) studied this question.