Electrical networks on $n$-simplex fractals

TL;DR

This paper investigates the asymptotic behavior of electrical networks on $n$-simplex fractals, revealing eigenspace structures and a new anisotropy exponent that characterizes anisotropic properties near the isotropic point.

Contribution

It introduces a detailed analysis of the decimation map for admittance networks on $n$-simplex fractals, including eigenspace characterization and a novel anisotropy exponent.

Findings

Eigenspaces near the isotropic point are always three for $n \,\geq\, 4$.

A new anisotropy exponent related to the third eigenspace is identified.

The anisotropy exponent exhibits a crossover between two logarithmic expressions.

Abstract

The decimation map $\mathcal{D}$ for a network of admittances on an $n$-simplex lattice fractal is studied. The asymptotic behaviour of $\mathcal{D}$ for large-size fractals is examined. It is found that in the vicinity of the isotropic point the eigenspaces of the linearized map are always three for $n \geq 4$; they are given a characterization in terms of graph theory. A new anisotropy exponent, related to the third eigenspace, is found, with a value crossing over from $\ln[(n+2)/3]/\ln 2$ to $\ln[(n+2)^3/n(n+1)^2]/\ln 2$.