Efficiency of the Incomplete Enumeration algorithm for Monte-Carlo simulation of linear and branched polymers

TL;DR

This paper analyzes the efficiency of the incomplete enumeration algorithm in Monte-Carlo simulations of linear and branched polymers, revealing different scaling behaviors and providing numerical evidence for a specific exponent in the branched case.

Contribution

It demonstrates a qualitative difference in algorithm efficiency between linear and branched polymers and identifies a specific scaling exponent for branched polymers.

Findings

Linear polymers: time scales as n^2

Branched polymers: time scales as exp(c n^{1/3})

Numerical evidence supports the exponent 1/3 for binary trees

Abstract

We study the efficiency of the incomplete enumeration algorithm for linear and branched polymers. There is a qualitative difference in the efficiency in these two cases. The average time to generate an independent sample of $n$ sites for large $n$ varies as $n^2$ for linear polymers, but as $exp(c n^{\alpha})$ for branched (undirected and directed) polymers, where $0<\alpha<1$. On the binary tree, our numerical studies for $n$ of order $10^4$ gives $\alpha = 0.333 \pm 0.005$. We argue that $\alpha=1/3$ exactly in this case.