Exotic trees
Jul, 2002
19 pages
Published in:
- Phys.Rev.E 67 (2003) 026105
e-Print:
- cond-mat/0207459 [cond-mat]
Report number:
- BI-TP-2002-15,
- TPJU-13-2002
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Abstract: (arXiv)
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.Note:
- 33 pages, 6 eps figures Report-no: BI-TP 2002/15, TPJU - 13/2002
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