# The average inter-crossing number of equilateral random walks and polygons

### Details

Serval ID

serval:BIB_31F454720298

Type

**Article**: article from journal or magazin.

Collection

Publications

Fund

Title

The average inter-crossing number of equilateral random walks and polygons

Journal

Journal of Physics A: Mathematical and General

Publication state

Published

Issued date

2005

Volume

38

Number

35

Pages

7601-7616

Language

english

Abstract

Abstract. In this paper, we study the average inter-crossing number between two random walks and two random polygons in the three-dimensional space. The random walks and polygons in this paper are the so-called equilateral random walks and polygons in which each segment of the walk or polygon is of unit length. We show that the mean average inter-crossing number ICN between two equilateral random walks of the same length n is approximately linear in terms of n and we were able to determine the prefactor of the linear term, which is . In the case of two random polygons of length n, the mean average inter-crossing number ICN is also linear, but the prefactor of the linear term is different from that of the random walks. These approximations apply when the starting points of the random walks and polygons are of a distance ρ apart and ρ is small compared to n. We propose a fitting model that would capture the theoretical asymptotic behaviour of the mean average ICN for large values of ρ. Our simulation result shows that the model in fact works very well for the entire range of ρ. We also study the mean ICN between two equilateral random walks and polygons of different lengths. An interesting result is that even if one random walk (polygon) has a fixed length, the mean average ICN between the two random walks (polygons) would still approach infinity if the length of the other random walk (polygon) approached infinity. The data provided by our simulations match our theoretical predictions very well.

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Create date

24/01/2008 11:36

Last modification date

03/03/2018 15:41