Exotic trees

Abstract

We discuss the scaling properties of free branched polymers. The scaling behaviour of the model is classified by the Hausdorff dimensions for the internal geometry: d_L and d_H, and for the external one: D_L and D_H. The dimensions d_H and D_H characterize the behaviour for long distances while d_L and D_L for short distances. We show that the internal Hausdorff dimension is d_L=2 for generic and scale-free trees, contrary to d_H which is known be equal two for generic trees and to vary between two and infinity for scale-free trees. We show that the external Hausdorff dimension D_H is directly related to the internal one as D_H = \alpha d_H, where \alpha is the stability index of the embedding weights for the nearest-vertex interactions. The index is \alpha=2 for weights from the gaussian domain of attraction and 0<\alpha <2 for those from the L\'evy domain of attraction. If the dimension D of the target space is larger than D_H one finds D_L=D_H, or otherwise D_L=D. The latter result means that the fractal structure cannot develop in a target space which has too low dimension.