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, 2≤b≤∞. The fractal dimension varies from log₂ 3 to 2 as b is varied from 2 to ∞. We find that for all b≥3, A_n varies as λⁿ exp(bn^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, 3≤b≤∞. This generalizes the earlier results of Knezevic and Vannimenus for b=3 [Phys. Rev B 35, 4988 (1987)].