Distribution of the distance between opposite nodes of random polygons with a fixed knot

TL;DR

This study numerically analyzes the distribution of distances between opposite nodes in random polygons with fixed knots, revealing Gaussian-like behavior and universal scaling properties across different knot types.

Contribution

It introduces a numerical analysis of distance distributions in knotted polygons and demonstrates universal scaling and Gaussian behavior for various knot types.

Findings

Distribution fits self-avoiding walk scaling form

Re-scaled distributions become universal Gaussian

Introduces a fitting formula for gyration radius distribution

Abstract

We examine numerically the distribution function $f_K(r)$ of distance $r$ between opposite polygonal nodes for random polygons of $N$ nodes with a fixed knot type $K$. Here we consider three knots such as $\emptyset$, $3_1$ and $3_1 \sharp 3_1$. In a wide range of $r$, the shape of $f_K(r)$ is well fitted by the scaling form of self-avoiding walks. The fit yields the Gaussian exponents $\nu_K = {1 \over 2}$ and $\gamma_K = 1$. Furthermore, if we re-scale the intersegment distance $r$ by the average size $R_K$ of random polygons of knot $K$, the distribution function of the variable $r/R_K$ should become the same Gaussian distribution for any large value of $N$ and any knot $K$. We also introduce a fitting formula to the distribution $g_K(R)$ of gyration radius $R$ for random polygons under some topological constraint $K$.