Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions - art. no. 056115

We show that in the loop-erased random-walk problem, the exponent character izing the probability distribution of areas of erased loops is superunivers al. In d dimensions, the probability that the erased loop has an area A var ies as A(-2) for large A, independent of d, for 2 less than or equal to d l ess than or equal to 4. We estimate the exponents characterizing the distri bution of perimeters and areas of erased loops in d = 2 and 3 by large-scal e Monte Carlo simulations. Our estimate of the fractal dimension z in two d imensions 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 the du rations of avalanches in the three-dimensional Abelian sandpile model is de termined from this by using scaling relations.