Branched Polymers on the Given-Mandelbrot family of fractals

TL;DR

This paper investigates the configurations of branched polymers on the Given-Mandelbrot fractals, deriving exact formulas for their growth and size exponents across a family of fractals with varying dimensions.

Contribution

It provides the first exact determination of the exponents governing branched polymer configurations on a family of fractals, generalizing previous specific cases.

Findings

A_n grows as λ^n exp(b n^ψ) for all b > 2

Exact values of the exponent ψ and size exponent ν are obtained

Results extend earlier work from specific to general fractal families

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 $ \lambda^n exp(b n ^{\psi})$, where $\lambda$ and $b$ are some constants, and $ 0 < \psi <1$. We determine the exponent $\psi$, and the size exponent $\nu$ (average diameter of polymer varies as $n^\nu$), exactly for all b > 2. This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev {\bf B 35} (1987) 4988].