POP approximation to the spectral dimension of dual three-dimensional Sierpinski carpets

The spectral dimension d of a network governs the massless singularity of a free field and the asymptotic behaviour of the diffusion on the network. Approximate values of d for two types of three-dimensional generalisations of the dual Sierpinski carpet are obtained using the POP approximation method. For one of them which is generated by a cube of side length three, with one block at the centre taken away, the value obtained is d(POP)=2 log 26/log(884/93) approximately=2.89. For the other one with seven cubes taken away, d(POP)=2 log 20/log (40/3) approximately=2.31. The algorithm of the POP method is explained. The results for 2D symmetric dual Sierpinski-type carpets are also reported.