Branched Polymers on the Given-Mandelbrot family of fractals

Abstract

We study the average number A_n per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b, b > 1. The fractal dimension varies from $ log_{_2} 3$ to 2 as b is varied from 2 to infinity. We find that for all b > 2, A_n varies as $ λ^n exp(b n ^ψ)$, where $λ$ and $b$ are some constants, and $ 0 < ψ<1$. We determine the exponent $ψ$, and the size exponent $ν$ (average diameter of polymer varies as $n^ν$), exactly for all b > 2. This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev {\bf B 35} (1987) 4988].