New approximate renormalization method on fractals

Self-organized stiffness in regular fractal polymer structures

We investigated membrane-like polymer structures of fractal connectivity such as Sierpinski gaskets and Sierpinski carpets applying the bond fluctuation model in three dimensions. Without excluded volume (phantom), both polymeric fractals obey Gaussian elasticity on larger scales determined by their spectral dimension. On the other hand, the swelling effect due to excluded volume is rather distinct between the two polymeric fractals: Self-avoiding Sierpinski gaskets can be described using a Flory-type mean-field argument. Sierpinski carpets having a spectral dimension closer to perfect membranes are significantly more strongly swollen than predicted. Based on our simulation results it cannot be excluded that Sierpinski carpets in athermal solvent show a flat phase on larger scales. We tested the self-consistency of Flory predictions using a virial expansion to higher orders. From this we conclude that the third virial coefficient contributes marginally to Sierpinski gaskets, but higher order virial coefficients are relevant for Sierpinski carpets.

On the resistance of the Sierpiński carpet

Let F n be the n th stage in the construction of the Sierpiński carpet. Let R n be the electrical resistance of F n when the left and right sides are each short-circuited, and a voltage is applied between them. We prove that there exists a constant ρ such that ¼ ρ n ≼ R n ≼ 4 ρ n . The motivation for this result came from the problem of establishing ( a ) the existence and ( b ) the value of the ‘spectral dimension’ of the Sierpiński carpet. In this and a subsequent paper, we settle ( a ) and give bounds for ( b ).