We will analyze shape and knotting in two polymer models: ideal rings and off-lattice self avoiding walks.

To describe the scaling and local shape of ideal rings, we will present two characteristics focusing on the shape and contribution of subsegments: the squared radius of gyration of subsegments and the squared internal end to end distance, which we define 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 corresponding values for 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.

We will then describe a new algorithm, which we refer to as the reflection method, to generate random walks of specified thickness in R 3 and prove that our method is ergodic. The data resulting from our implementation of this method will allows us to describe the complex relationship between the presence and nature of knotting and size, thickness and shape of the walk. We will expand on the current understanding of excluded volume by analyzing how scaling of the squared radius of gyration is effected by the introduction of thickness. We will also examine the profound effect of thickness on knotting in open chains.