  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.
