Abstract
We present a family of exact fractals with a wide range of fractal and fracton dimensionalities. This includes the case of the fracton dimensionality of 2, which is critical for diffusion. This is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal. These fractals include the cases of finite and infinite ramification characterized by a ramification exponentp. The infinite ramification makes the problem of percolation on these lattices a nontrivial one. We give numerical evidence for a percolation transition on these fractals. This transition is tudied by a real-space renormalization group technique on lattices with fractal dimensionality ¯d between 1 and 2. The critical exponents for percolation depend strongly on the geometry of the fractals.
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Havlin, S., Ben-Avraham, D. & Movshovitz, D. Percolation on infinitely ramified fractals. J Stat Phys 36, 831–841 (1984). https://doi.org/10.1007/BF01012943
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DOI: https://doi.org/10.1007/BF01012943