Fractal structure of Ising and Potts clusters: Exact results

It is shown that previously defined clusters, which give a geometrical description of the fluctuations in the q-state Potts model, at criticality have a fractal structure made of links and blobs as in percolation. Using the mapping from the Potts model to the Coulomb gas it is found that the fractal dimension of the links or red bonds $D_R$ is given by 5/4, 3/4, 13/24, 7/20, 0, while the fractal dimension of the external hull ${\mathit{D}}_{\mathit{h}}$ is given by 2, 7/4, 5/3, 8/5, 3/2, for q=0,1,2,3,4. A model originally introduced by Mandelbrot and Given for percolation clusters is found to correctly describe the fractal structure of the Potts clusters.