Dimensional Interpolation for Random Walk

TL;DR

This paper introduces a dimensional interpolation method to accurately estimate the shapes of random walks in 2D and 3D, achieving about 2% error compared to numerical results, and can be applied to other properties.

Contribution

A simple and accurate dimensional interpolation formula for random walk shapes at D=2 and D=3 based on known solutions at D=1 and D=infinity.

Findings

Radi of gyration estimates with ~2% error

Asphericity calculations match simulations

Method applicable to other random walk properties

Abstract

We employ a simple and accurate dimensional interpolation formula for the shapes of random walks at $D=3$ and $D=2$ based on the analytically known solutions at both limits $D=\infty$ and $D=1$. The results obtained for the radii of gyration of an arbitrary shaped object are about $2\%$ error compared with accurate numerical results at $D = 3$ and $D = 2$. We also calculated the asphericity for a three-dimensional random walk using the dimensional interpolation formula. Result agrees very well with the numerically simulated result. The method is general and can be used to estimate other properties of random walks.