# Off-Lattice Random Walks with Excluded Volume: A New Method of Generation, Proof of Ergodicity and Numerical Results

## TL;DR

This paper introduces a new ergodic algorithm for generating off-lattice random walks with variable thickness, analyzes their scaling properties, and investigates how thickness influences knotting probability and chain size.

## Contribution

The authors present a novel reflection method for generating thick off-lattice random walks and prove its ergodicity, expanding understanding of excluded volume effects on polymer knotting and size.

## Key findings

- Scaling of squared radius of gyration with length and thickness confirmed.
- Increasing thickness significantly reduces knotting probability in open chains.
- Thickness impacts the size and knotting behavior of random walks, with quantifiable effects.

## Abstract

We describe a new algorithm, the reflection method, to generate off-lattice random walks of specified, though arbitrarily large, thickness in $\mathbb{R}^3$ and prove that our method is ergodic on the space of thick walks. The data resulting from our implementation of this method is consistent with the scaling of the squared radius of gyration of random walks, with no thickness constraint. Based on this, we use the data to describe the complex relationship between the presence and nature of knotting and size, thickness and shape of the random walk. We extend the current understanding of excluded volume by expanding the range of analysis of how the squared radius of gyration scales with length and thickness. We also examine the profound effect of thickness on knotting in open chains. We will quantify how thickness effects the size of thick open chains, calculating the growth exponent for squared radius of gyration as a function of thickness. We will also show that for radius $r \leq 0.4$, increasing thickness by 0.1 decreases the probability of knot formation by 50% or more.