Clayton Shonkwiler,  CSU Math

A New Algorithm for Sampling Closed Equilateral Random Walks

An equilateral random walk in Euclidean 3-space is formed by sampling steps uniformly on the unit sphere and summing the steps to form a piecewise-linear path. Such walks arise in many contexts, including in the numerical simulation and probabilistic analysis of polymers in solution. Simulation and analysis are relatively straightforward when the steps are independent, but independence fails and things get much trickier when the topology of the walk is constrained--for example, when modeling "ring" polymers which form closed loops. In this talk, I will describe an unbiased sampling algorithm for equilateral random walks which form a closed loop. The justification for the fact that it is unbiased uses the well-known toric symplectic structure on the moduli space of closed walks, but the algorithm itself is purely combinatorial. I will also mention some tantalizing but still-mysterious connections to Catalan numbers, integer partitions, and other combinatorial gadgets. This is joint work with Jason Cantarella (University of Georgia) and Erica Uehara (Ochanomizu University, Tokyo).