Branched Polymers on the Given-Mandelbrot family of fractals

TL;DR

This paper analyzes branched polymers on the Given-Mandelbrot family of fractals using exact real-space renormalization. It shows that for all $b>2$ the number of configurations grows as $A_n \sim \lambda^n \exp(b\,n^{\psi})$, with $\nu$ and $\psi$ determined exactly for every $b\ge3$, and that $G(x)$ has an essential singularity driving stretched-exponential corrections. The dominant-term analysis reduces the complex polynomial recursion to a tractable framework yielding exact expressions for the exponents in terms of $b$, revealing a geometry-induced crossover from Euclidean to fractal behavior and generalizing prior results for smaller $b$. These results illuminate how fractal geometry influences polymer scaling and provide a methodological template for handling similarly inhomogeneous deterministic lattices. Open questions include extensions to collapse transitions, quenched averaging, and the precise nature of transverse exponents like $z$.

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].

Figures (8)

• Figure 1: The recursive construction of the Given-Mandelbrot fractal for $b =4$. (a) The graph of first order riangle.(b) the graph of a $(r +1)$ order triangle, formed by joining $b(b+1)/2$$r$-th order triangles shown as shaded triangles here.
• Figure 2: The graph of a $5$-th order triangle for $b = 2$.
• Figure 3: Definition of the restricted partition functions $A^{(r)}, B^{(r)}, C^{(r)},D^{(r)}, E^{(r)}$ and $F^{(r)}$.
• Figure 4: Local modification of polymer confugurations to increase its weight. The weight increases is going from configuration $C_1$ to $C_2$, and from $C_3$ to $C_4$.
• Figure 5: Some examples of configurations where extending the polymer configuration is not favorable. Here $C_5$ has higher weight than $C_6$, and $C_7$ has higher weight than $C_8$.