Fractal dimension of backbone of Eden trees.

We relate the fractal dimension of the backbone, and the spectral dimension of Eden trees to the dynamical exponent z. In two dimensions, it gives fractal dimension of backbone equal to 4/3 and spectral dimension of trees equal to 5/4. In three dimensions, it provides us a new way to estimate z numerically. We get z = 1.617± 0.004. Dense branching patterns are found in many different physical situations in nature e.g., coral reefs, river networks, collapsed phase of branched polymers, very slowly evaporated films of sugar dissolved in water [1-3]. In all these systems, the Hausdorff dimension of the structure is equal to that of the embedding space but the detailed structure is different depending on the different physical processes involved. The Eden model has been studied a lot in the last decade, mainly for the surface properties [4]. Eden trees [5,6] are simple theoretical model of dense branching structures. In [5], it was argued that classical diffusion on Eden trees is anomalous because of trapping in dead end branches, and the root mean square deviation of a random walker on the tree increases with time as t, where the exponent x does not satisfy the usual relation x = d̃/2d̄, where d̃ is the spectral dimension, and d̄ is the (Hausdorff) fractal dimension of the lattice. Dhar and Ramaswamy expressed the exponents x and d̃ in terms of an exponent θ related to the fractal dimension of the backbone of the trees [5]. Using a different method of analysis, and somewhat larger simulations, Nakanishi and Herrmann [6] also calculated these exponents. However, these exponents have not been determined analytically so far. In this Rapid Communication, we show that the backbone exponent of the Eden trees can be related to the dynamical exponent of KPZ model, and show that in two dimensions,θ = 1/3, d̃ = 5/4 and x = 3/8. In three dimensions, our method gives us a new way to determine the dynamical exponent numerically. The numerical determined value is z = 1.617± 0.004. We shall consider Bond-Eden Trees (BET) in this report. The model is defined in