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

We examine numerically the distribution function f K (r) of the distance r between opposite polygonal nodes for random polygons with a fixed knot type K. Here we consider some knots such as ∅, 3 1 and 3 1 ♯3 1. In a wide range of r, the shape of f K (r) is well fit to the scaling form [1] derived from the field theory for self-avoiding walks. The fit yields the exponents ν K = 1 2 and γ K = 1, which are consistent with the exponents of the Gaussian distribution. We also introduce a fitting formula to the distribution g K (R) of the gyration radius R for random polygons under some topological constraint K. It is significant for the form of the distribution whether random polygons have a topological constraint or not.