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

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 $\varnothing$, $3_1$ and $3_1♯3_1$. 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 ${\nu }_K=\frac{1}{2}$ 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 $K$.

15 pages, 4 figures, 2 tables