The mixing time of the Dikin walk in a polytope - A simple proof

Abstract. Sampling points from the uniform distribution on a polytope is a well-studied problem, and is an important ingredient in several computational tasks involving polytopes, such as volume estimation. This is achieved by setting up a random walk inside the polytope, with its stationary distribution being uniform in the interior of the polytope. Kannan-Narayanan [6] and Narayanan [9] proposed the Dikin walk based on interior point methods, where the next point is sampled, roughly, from the DIkin ellipsoid at the current point. In this paper, we give a simple proof of the mixing time of the Dikin walk, using well-known properties of Gaussians, and concentration of Gaussian polynomials.