Asymptotic scaling behavior of self-avoiding walks on critical percolation clusters

TL;DR

This paper introduces a new enumeration method to analyze self-avoiding walks on critical percolation clusters, revealing a smaller scaling exponent and challenging existing scaling laws with extensive data.

Contribution

The authors develop an exact enumeration technique exploiting fractal structures, enabling analysis of much longer walks and providing new insights into scaling behavior.

Findings

The scaling exponent ν is smaller than previously estimated.

ν appears consistent on backbones and full clusters.

Proposes an alternative scaling law for the number of conformations.

Abstract

We study self-avoiding walks on three-dimensional critical percolation clusters using a new exact enumeration method. It overcomes the exponential increase in computation time by exploiting the clusters' fractal nature. We enumerate walks of over $10^4$ steps, far more than has ever been possible. The scaling exponent $\nu$ for the end-to-end distance turns out to be smaller than previously thought and appears to be the same on the backbones as on full clusters. We find strong evidence against the widely assumed scaling law for the number of conformations and propose an alternative, which perfectly fits our data.