Position and Orientation Distributions for Non-Reversal Random Walks using Space-Group Fourier Transforms

• Published in: Journal of algebraic statistics Vol. 1; no. 1; p. 27
• Main Authors: Skliros, Aris, Park, Wooram, Chirikjian, Gregory S
• Format: Journal Article
• Language: English
• Published: Turkey 01.01.2010
• ISSN: 1309-3452; 1309-3452
• DOI: 10.18409/jas.v1i1.6

This paper presents an efficient group-theoretic approach for computing the statistics of non-reversal random walks (NRRW) on lattices. These framed walks evolve on proper crystallographic space groups. In a previous paper we introduced a convolution method for computing the statistics of NRRWs in which the convolution product is defined relative to the space-group operation. Here we use the corresponding concept of the fast Fourier transform for functions on crystallographic space groups together with a non-Abelian version of the convolution theorem. We develop the theory behind this technique and present numerical results for two-dimensional and three-dimensional lattices (square, cubic and diamond). In order to verify our results, the statistics of the end-to-end distance and the probability of ring closure are calculated and compared with results obtained in the literature for the random walks for which closed-form expressions exist.