Asymptotic form of the spectral dimension of the Sierpinski gasket type of fractals

Abstract

The authors have studied the spectral dimension d48T of an infinite class of fractals. The first member (b=2) of the class is the two-dimensional Sierpinski gasket, while the last member (b= infinity ) appears to be a wedge of the ordinary triangular lattice. By studying the electric resistance of the fractals they have been able to calculate exact values of d for the first 200 members of the class. An analysis of the obtained data reveals that for large b the spectral dimension should approach the upper limit of 2 according to the formula d approximately=2-constant (ln b)beta, where beta is not larger than one. This result implies, among other things, that the scaling exponents of the resistivity and diffusion constant should logarithmically vanish at the fractal-lattice crossover.