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

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

2001-07-12

arxiv.org/abs/cond-mat/0107246v1

Phys. Rev E 65 (2002) 031108

Physics

Condensed Matter

Statistical Mechanics

11 pages, 1 table, 10 included figures, revtex

Scientific paper

10.1103/PhysRevE.65.031108

We have studied the probability distribution of the perimeter and the area of the k-th 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 k-th 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 <= 20 to determine the probability that no loop of size greater than l (ell) is erased. We show that correlations between loops have to be taken into account to describe the average size of the k-th largest erased-loops. We propose a one-dimensional Levy walk model which takes care of these correlations. The simulations of this simpler model compare very well with the simulations of the original problem.

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