Exotic trees
Abstract
We discuss the scaling properties of free branched polymers. The scaling behavior 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 behavior 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 scalefree trees, contrary to d_{H}, which is known be equal to 2 for generic trees and to vary between 2 and ∞ for scalefree trees. We show that the external Hausdorff dimension D_{H} is directly related to the internal one as D_{H}=αd_{H}, where α is the stability index of the embedding weights for the nearestvertex 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 that has too low dimension.
 Publication:

Physical Review E
 Pub Date:
 February 2003
 DOI:
 10.1103/PhysRevE.67.026105
 arXiv:
 arXiv:condmat/0207459
 Bibcode:
 2003PhRvE..67b6105B
 Keywords:

 05.40.a;
 64.60.i;
 Fluctuation phenomena random processes noise and Brownian motion;
 General studies of phase transitions;
 Condensed Matter;
 High Energy Physics  Lattice;
 High Energy Physics  Theory
 EPrint:
 33 pages, 6 eps figures