The Geometry of Constrained Randomwalks and an Application to Frame Theory

Abstract

Random walks in ${{\mathbb{R}}^3}$ are classical objects in geometric probability which have, over the last 70 years, been rather successfully used as models of polymers in solution. Modifying the theory to apply to topologically nontrivial polymers, such as ring polymers, has proven challenging, but several recent breakthroughs have been made by thinking of random walks as points in some nice conformation space and then exploiting the geometry of the space. Using tools from symplectic geometry, this approach yields a fast algorithm for sampling loop random walks. Such walks can be lifted via the Hopf map to finite unit norm tight frames (FUNTFs) in ${{\mathbb{C}}^2}$, producing an algorithm for randomly sampling FUNTFs in ${{\mathbb{C}}^2}$ as well as a mechanism for searching for FUNTFs with nice properties. In general, symplectic geometry seems like a promising tool for understanding the space of FUNTFs in ${{\mathbb{C}}^d}$ for any d.