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Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions

Himanshu Agrawal·Deepak Dhar
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Abstract

We show that in the loop-erased random walk problem, the exponent characterizing probability distribution of areas of erased loops is superuniversal. In d-dimensions, the probability that the erased loop has an area A varies as A^{-2} for large A, independent of d, for 2 <= d <= 4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d = 2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two-dimensions is consistent with the known exact value 5/4. In three-dimensions, we get z = 1.6183 +- 0.0004. The exponent for the distribution of durations of avalanche in the three-dimensional abelian sandpile model is determined from this by using scaling relations.

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BibTeXRIS

Himanshu Agrawal, Deepak Dhar. 2000-12-07. Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions. https://doi.org/10.1103/physreve.63.056115

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