Exotic trees

TL;DR

This paper analyzes the scaling properties of free branched polymers, classifying their internal and external geometries through Hausdorff dimensions, and explores how these dimensions relate to the embedding space and weight distributions.

Contribution

It provides a detailed classification of the scaling behavior of branched polymers using Hausdorff dimensions and relates external and internal geometries to the stability index of weights.

Findings

Internal Hausdorff dimension d_L=2 for generic and scale-free trees.

External Hausdorff dimension D_H is proportional to internal dimension d_H, D_H = lpha d_H.

Fractal structure depends on the target space dimension D, with D_L=D_H if D > D_H, otherwise D_L=D.

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.