A random walk in 3-space is a classical object in geometric probability, given by choosing a direction at random, taking a step, and repeating n times. Since the directions of the steps are independent random variables, many problems about random walks can be solved analytically. Random walks are used in polymer physics to describe the possible configurations of a "linear" polymer composed of n identical monomers.

However, many biological polymers are "ring" polymers; to model these we must add the constraint that the random walk is closed, meaning that it returns to its starting point after n steps. In this situation the directions of the steps are no longer independent and closed-form solutions become much harder to come by.

In this talk I will describe some applications of the symplectic structure on polygon spaces to the study of closed random walks. The symplectic structure clarifies the situation considerably, leading to new algorithms as well as explicit computations of probabilities which had previously seemed far out of reach. This is joint work with Jason Cantarella (University of Georgia).