Characteristics of shape and knotting in ideal rings

We present two descriptions of the the local scaling and shape of ideal rings, primarily featuring subsegments. Our focus will be the squared radius of gyration of subsegments and the squared internal end to end distance, defined to be the average squared distance between vertices $k$ edges apart. We calculate the exact averages of these values over the space of all such ideal rings, not just a calculation of the order of these averages, and compare these to the equivalent values in open chains. This comparison will show that the structure of ideal rings is similar to that of ideal chains for only exceedingly short lengths. These results will be corroborated by numerical experiments. They will be used to analyze the convergence of our generation method and the effect of knotting on these characteristics of shape.