Ring Polymer chains, where the last monomer is attached to the first, are the subject of extensive research in the recent years, both in the context of biological systems, such DNA plasmids and the study of chromosomal territories in the nucleus of eukaryotic cells, as well as chemical systems.

The size and shape distributions of these polymers have direct impact on their packing properties, reaction rates and the characteristics of their diffusion, among others.  Cutting-edge experiments are now able to probe the properties of single such molecules and their fluctuations, one frame at a time. One parameter which is mainly used for characterizing the size and shape of ring polymers is the Radius of Gyration, Rg. Other commonly used observables such as the end-to-end distance, are irrelevant for rings. Using new tools developed in the stochastic processes community we recognized a new parameter which describes the fluctuations of these polymers, and find its distribution for chemically identical rings. In addition to the the non-interacting chain, we consider locally interacting rings and self-avoiding rings with exluded volume.