Hypersimplices are the slices of cubes perpendicular to the vector (1,1, ... ,1). The problem of computing a mixed volume involving these polytopes turns out to be equivalent to computing the probability that a certain random walk of many points on a circle ends up in a particular position. This enables one to compute many of these mixed volumes quickly. Any mixed volume of polytopes can be phrased in this way but in general will lead to a random walk in a larger space and with many final positions instead of just one.

This is joint work with Einar Steingrímsson.