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 = αd_H, where αis the stability index of the embedding weights for the nearest-vertex interactions. The index is α=2 for weights from the gaussian domain of attraction and 0<α<2 for those from the Lévy 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.

33 pages, 6 eps figures

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Cited by in corpus (6)

• The effects of degree correlations on network topologies and robustness
• Polyakov Lines in Yang-Mills Matrix Models
• The statistical geometry of scale-free random trees
• Ageing of complex networks
• Appearance of vertices of infinite order in a model of random trees
• Long-range disassortative correlations in generic random trees