Agrawal H, Dhar D
Theoretical Physics Group, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400 005, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 May;63(5 Pt 2):056115. doi: 10.1103/PhysRevE.63.056115. Epub 2001 Apr 19.
We show that in the loop-erased random-walk problem, the exponent characterizing the 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< or =d< or =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 the durations of avalanches in the three-dimensional Abelian sandpile model is determined from this by using scaling relations.
我们表明，在圈擦除随机游走问题中，表征擦除圈面积概率分布的指数是超普适的。在(d)维空间中，对于(2\leq d\leq4)，当(A)很大时，擦除圈面积为(A)的概率随(A^{-2})变化，与(d)无关。我们通过大规模蒙特卡罗模拟估计了二维和三维中表征擦除圈周长和面积分布的指数。我们对二维分形维数(z)的估计与已知的精确值(5/4)一致。在三维中，我们得到(z = 1.6183\pm0.0004)。通过使用标度关系，由此确定了三维阿贝尔沙堆模型中雪崩持续时间分布的指数。
