PMID: 11909030Mar 23, 2002Paper

Probability distribution of the sizes of the largest erased loops in loop-erased random walks

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
H Agrawal, D Dhar

Abstract

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.

References

Jul 27, 1987·Physical Review Letters·P BakK Wiesenfeld
Dec 23, 1996·Physical Review Letters·A R Conway, A J Guttmann
Oct 25, 2000·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·D V Ktitarev B Priezzhev V
Nov 23, 2000·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics· De Menech M, A L Stella
Jun 21, 2001·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·H Agrawal, D Dhar

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