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

We examine numerically the distribution function fK(r) of the 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 , 31 and 31 sharp 31. In a wide range of r, the shape of fK(r) is well fitted by the scaling form [1] of self-avoiding walks. The fit yields the Gaussian exponents νK=1/2 and γK = 1. Furthermore, if we re-scale the intersegment distance r by the average size RK of random polygons of knot K, the distribution function of the variable r/RK 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 gK(R) of gyration radius R for random polygons under some topological constraint K