  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.
