Random walks on hierarchical lattices at the percolation threshold

Random walks on hierarchical lattices are considered at the percolation threshold. Real-space renormalisation methods are employed to obtain the exact fractal dimension of the incipient infinite cluster (IIC), d f , and the fractal dimension of the effective one-dimensional resistance length of the IIC, d r , for some hierarchical lattices. The recent conjecture of Aharony and Stauffer does not hold for diamond and Wheatstone bridge hierarchical lattices.