Spectral dimension of a wire network

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Abstract
By a rigorous analysis for diffusion on a wire network the spectral dimension of a fractal is shown to be independent of the local structure. It is discussed that diffusion does not occur when the lattice spacing tends to zero on such a fractal that the spectral dimension is less than the Hausdorff dimension if a free field on the fractal exhibits a certain long distance behaviour. For a Sierpinski carpet, the spectral dimension is evaluated within bond-moving approximation (Migdal-Kadanov renormalisation). As a result, the author obtains a value smaller than the Hausdorff dimension.