The linking number and the writhe of uniform random walks and polygons in confined spaces

TL;DR

This paper extends the concepts of writhe, self-linking, and linking number to open chains and studies their average behavior in confined spaces, revealing quadratic and square-root scaling laws through theoretical analysis and numerical simulations.

Contribution

It introduces the extension of linking and writhe to open chains and establishes their asymptotic behavior in confined spaces, supported by numerical evidence.

Findings

Mean squared linking number, writhe, and self-linking number scale as O(n^2).

Mean absolute linking number between a fixed curve and a random walk scales as O(√n).

Mean absolute linking number between two random walks scales as O(n).

Abstract

Random walks and polygons are used to model polymers. In this paper we consider the extension of writhe, self-linking number and linking number to open chains. We then study the average writhe, self-linking and linking number of random walks and polygons over the space of configurations as a function of their length. We show that the mean squared linking number, the mean squared writhe and the mean squared self-linking number of oriented uniform random walks or polygons of length $n$, in a convex confined space, are of the form $O(n^2)$. Moreover, for a fixed simple closed curve in a convex confined space, we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of $n$ edges is of the form $O(\sqrt{n})$. Our numerical studies confirm those results. They also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons, of $n$ edges each, is of the form O(n). Equilateral random walks and polygons are used to model polymers in $\theta$-conditions. We use numerical simulations to investigate how the self-linking and linking number of equilateral random walks scale with their length.