The topological Hausdorff dimension and transport properties of Sierpiński carpets

In this Letter, the analytical expression of topological Hausdorff dimension DtH is derived for some kinds of infinitely ramified Sierpiński carpets. Furthermore, we deduce that the Hausdorff dimension of the union of all self-avoiding paths admitted on the infinitely ramified Sierpiński carpet has the Hausdorff dimension DHsa=DtH. We also put forward a phenomenological relation for the fractal dimension of the random walk on the infinitely ramified Sierpiński carpet. The effects of fractal attributes on the transport properties are highlighted. Possible applications of the developed tools are briefly outlined.