Restoration of isotropy on fractals

TL;DR

This paper demonstrates a universal phenomenon where macroscopic isotropy is restored in fractals with microscopic anisotropy across various physical systems, unlike in translationally invariant spaces.

Contribution

It introduces a new type of homogenization process specific to fractals, highlighting a universal mechanism absent in translationally invariant spaces.

Findings

Restoration of isotropy observed in fractals with microscopic anisotropy

Universal across diffusions, random walks, resistor networks, and Gaussian fields

Distinct from behavior in translationally invariant spaces

Abstract

We report a new type of restoration of macroscopic isotropy (homogenization) in fractals with microscopic anisotropy. The phenomenon is observed in various physical setups, including diffusions, random walks, resistor networks, and Gaussian field theories. The mechanism is unique in that it is absent in spaces with translational invariance, while universal in that it is observed in a wide class of fractals.