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

E. Panagiotou, K. C. Millett, S. Lambropoulou

Published 2009-07-22Version 1

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.