Linking probabilities of off-lattice self-avoiding polygons and the effects of excluded volume

We evaluate numerically the probability of linking, i.e. the probability of a given pair of self-avoiding polygons (SAPs) being entangled and forming a nontrivial link type L. In the simulation we generate pairs of SAPs of N spherical segments of radius rd such that they have no overlaps among the segments and each of the SAPs has the trivial knot type. We evaluate the probability of a self-avoiding pair of SAPs forming a given link type L for various link types with fixed distance R between the centers of mass of the two SAPs. We define normalized distance r by where denotes the square root of the mean square radius of gyration of SAP of the trivial knot 01. We introduce formulae expressing the linking probability as a function of normalized distance r, which gives good fitting curves with respect to chi2 values. We also investigate the dependence of linking probabilities on the excluded-volume parameter rd and the number of segments, N. Quite interestingly, the graph of linking probability versus normalized distance r shows no N-dependence at a particular value of the excluded volume parameter, rd = 0.2.