Anomalous finite-size effects and canonical asymptotic behaviors for the mean-squared gyration radius of Gaussian random knots

Anomalously strong finite-size effects have been observed for the mean square radius of gyration $R^2_K$ of Gaussian random polygons with a fixed knot $K$ as a function of the number $N$ of polygonal nodes. Through computer simulations with $N < 2000$, we find for several knots that the gyration radius $R^2_K$ can be approximated by a power law: $R^2_K \sim N^{2 \nu_{K}^{eff}}$, where the effective exponents $\nu_{K}^{eff}$ for the knots are larger than 0.5 and less than 0.6. A crossover occurs for the gyration radius of the trivial knot, when $N$ is roughly equal to the characteristic length $N_c$ of random knotting. For the asymptotic behavior of $R^2_K$, however, we find that it is consistent with the standard one with the scaling exponent 0.5. Thus, although the strong finite-size effects of $R^2_K$ remain effective for extremely large values of $N$, they can be matched with the asymptotic behavior of random walks.