Agrawal Himanshu, Dhar Deepak
Department of Theoretical Physics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Mar;65(3 Pt 1):031108. doi: 10.1103/PhysRevE.65.031108. Epub 2002 Mar 1.
We have studied the probability distribution of the perimeter and the area of the kth largest erased loop in loop-erased random walks in two dimensions for k=1 to 3. For a random walk of N steps, for large N, the average value of the kth largest perimeter and area scales as N(5/8) and N, respectively. The behavior of the scaled distribution functions is determined for very large and very small arguments. We have used exact enumeration for N< or =20 to determine the probability that no loop of size greater than l is erased. We show that correlations between loops have to be taken into account to describe the average size of the kth largest erased loops. We propose a one-dimensional Levy walk model that takes care of these correlations. The simulations of this simpler model compare very well with the simulations of the original problem.
我们研究了二维环消随机游走中第(k)大的被消去环的周长和面积的概率分布，其中(k = 1)到(3)。对于(N)步的随机游走，当(N)很大时，第(k)大的周长和面积的平均值分别按(N^{5/8})和(N)缩放。确定了缩放后的分布函数在非常大的和非常小的自变量情况下的行为。对于(N\leq20)，我们使用精确枚举来确定大小大于(l)的环未被消去的概率。我们表明，为了描述第(k)大的被消去环的平均大小，必须考虑环之间的相关性。我们提出了一个一维列维游走模型来处理这些相关性。这个更简单模型的模拟结果与原始问题的模拟结果非常吻合。
