Superuniversal spectral dimension for dilute branched polymers?

Abstract

The authors introduce the problem of random walks on lattice animals (dilute branched polymers), and calculate the spectral dimension ds 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 ds=2da/dw where da is the fractal dimension of the lattice animal, and dw the fractal dimension of the random walk on the lattice animal. The results indicate that ds may be a superuniversal quantity, i.e. its value is independent of dimension. Moreover, the authors find ds to be close to 4/3 which is the spectral dimension of the largest percolation cluster at the percolation threshold pc as conjectured by Alexander and Orbach (1982). Since both lattice animals and the largest percolation cluster at pc have a homogeneous interior structure, the results suggest that the spectral dimension of all such fractals may equal 4/3.