Superuniversal spectral dimension for dilute branched polymers?

The authors introduce the problem of random walks on lattice animals (dilute branched polymers), and calculate the spectral dimension d s of animals in two and three dimensions by using a two-parameter position-space renormalisation group method. This problem, which is a generalisation of the problem of 'the ant in the labyrinth', has been independently proposed and investigated by Wilke et al. (1984). The spectral dimension is given by d s =2d a /d w where d a is the fractal dimension of the lattice animal, and d w the fractal dimension of the random walk on the lattice animal. The results indicate that d s may be a superuniversal quantity, i.e. its value is independent of dimension. Moreover, the authors find d s to be close to 4 / 3 which is the spectral dimension of the largest percolation cluster at the percolation threshold p c as conjectured by Alexander and Orbach (1982). Since both lattice animals and the largest percolation cluster at p c have a homogeneous interior structure, the results suggest that the spectral dimension of all such fractals may equal 4 / 3 .