Where two fractals meet: the scaling of a self-avoiding walk on a percolation cluster

TL;DR

This paper uses a field-theoretical renormalization group approach to analyze the scaling properties of self-avoiding walks on percolation clusters, providing an analytic expression for the critical exponent nu_p that matches numerical data across multiple dimensions.

Contribution

It reexamines a previous model to derive a more accurate analytic expression for the scaling exponent of self-avoiding walks on percolation clusters using renormalization group techniques.

Findings

Derived an explicit formula for the exponent nu_p in terms of epsilon.

The analytic results agree well with Monte Carlo and enumeration data.

Extended the understanding of multifractal properties of self-avoiding walks on percolation clusters.

Abstract

The scaling properties of self-avoiding walks on a d-dimensional diluted lattice at the percolation threshold are analyzed by a field-theoretical renormalization group approach. To this end we reconsider the model of Y. Meir and A. B. Harris (Phys. Rev. Lett. 63:2819 (1989)) and argue that via renormalization its multifractal properties are directly accessible. While the former first order perturbation did not agree with the results of other methods, we find that the asymptotic behavior of a self-avoiding walk on the percolation cluster is governed by the exponent nu_p=1/2 + epsilon/42 + 110epsilon^2/21^3, epsilon=6-d. This analytic result gives an accurate numeric description of the available MC and exact enumeration data in a wide range of dimensions 2<=d<=6.