Self-similarity and covered neighborhoods of fractals: A random walk test

D. Stauffer; A. Aharony; B. B. Mandelbrot

doi:10.1016/0378-4371(93)90076-G

Self-similarity and covered neighborhoods of fractals: A random walk test

D. Stauffer, A. Aharony, B. B. Mandelbrot

Research output: Contribution to journal › Article › peer-review

9 Scopus citations

Abstract

A strong version of the property of self-similarity is described, and it is shown that this property is satisfied by random walks on a simple cubic lattice. When each site visited by the walk is surrounded by a small cube, the total volume of these covering cubes depends on the cube size, the size of the region investigated, and the length of the walk. We find that for long walks at a fixed ratio of cube to region size the filling ratio is roughly constant.

Original language	English
Pages (from-to)	1-5
Number of pages	5
Journal	Physica A: Statistical Mechanics and its Applications
Volume	196
Issue number	1
DOIs	https://doi.org/10.1016/0378-4371(93)90076-G
State	Published - 15 May 1993
Externally published	Yes

ASJC Scopus subject areas

Statistics and Probability
Condensed Matter Physics

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title = "Self-similarity and covered neighborhoods of fractals: A random walk test",

abstract = "A strong version of the property of self-similarity is described, and it is shown that this property is satisfied by random walks on a simple cubic lattice. When each site visited by the walk is surrounded by a small cube, the total volume of these covering cubes depends on the cube size, the size of the region investigated, and the length of the walk. We find that for long walks at a fixed ratio of cube to region size the filling ratio is roughly constant.",

author = "D. Stauffer and A. Aharony and Mandelbrot, {B. B.}",

note = "Funding Information: We thank the German-Israeli foundation and BMFT grant 0326657D for partial support of this work, GMD St. Augustin and KFA Jiilich for use of their parallel i860 computers, and J. J/ickle for discussions.",

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AU - Aharony, A.

AU - Mandelbrot, B. B.

N1 - Funding Information: We thank the German-Israeli foundation and BMFT grant 0326657D for partial support of this work, GMD St. Augustin and KFA Jiilich for use of their parallel i860 computers, and J. J/ickle for discussions.

PY - 1993/5/15

Y1 - 1993/5/15

N2 - A strong version of the property of self-similarity is described, and it is shown that this property is satisfied by random walks on a simple cubic lattice. When each site visited by the walk is surrounded by a small cube, the total volume of these covering cubes depends on the cube size, the size of the region investigated, and the length of the walk. We find that for long walks at a fixed ratio of cube to region size the filling ratio is roughly constant.

AB - A strong version of the property of self-similarity is described, and it is shown that this property is satisfied by random walks on a simple cubic lattice. When each site visited by the walk is surrounded by a small cube, the total volume of these covering cubes depends on the cube size, the size of the region investigated, and the length of the walk. We find that for long walks at a fixed ratio of cube to region size the filling ratio is roughly constant.

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Self-similarity and covered neighborhoods of fractals: A random walk test

Abstract

ASJC Scopus subject areas

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