## Abstract

We study a kind of 'restoration of isotropy" on the pre-Sierpiński carpet. Let R^{x}_{n}(r) and R^{y}_{n}(r) be the effective resistances in the x and y directions, respectively, of the Sierpiński carpet at the n^{th} stage of its construction, if it is made of anisotropic material whose anisotropy is parametrized by the ratio of resistances for a unit square: r = R^{y}_{0} / R^{x}_{0}. We prove that isotropy is weakly restored asymptotically in the sense that for all sufficiently large n the ratio R^{y}_{n}(r) / R^{x}_{n}(r) is bounded by positive constants independent of r. The ratio decays exponentially fast when r ≫ 1. Furthermore, it is proved that the effective resistances asymptotically grow exponentially with an exponent equal to that found by Barlow and Bass for the isotropic case r = 1.

Original language | English |
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Pages (from-to) | 1-27 |

Number of pages | 27 |

Journal | Communications in Mathematical Physics |

Volume | 188 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1997 Jan 1 |

Externally published | Yes |

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics