Electrical networks on $n$-simplex fractals

Details Electrical networks on $n$-simplex fractals Electrical networks on $n$-simplex fractals

2007-09-27

arxiv.org/abs/0709.4360v1

Journal of Physics A 40, 12397-12408 (2007)

Physics

Condensed Matter

Disordered Systems and Neural Networks

14 pages, 8 figures

Scientific paper

10.1088/1751-8113/40/41/009

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 $\ln[(n+2)^3/n(n+1)^2]/\ln 2$.

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