Mixing time of an unaligned Gibbs sampler on the square

Abstract

The paper concerns a particular example of the Gibbs sampler and its mixing efficiency. Coordinates of a point are rerandomized in the unit square $[0,1{]}^2$ to approach a stationary distribution with density proportional to $\exp (-A^2(u-v{)}^2)$ for $(u,v)\in [0,1{]}^2$ with some large parameter $A$. Diaconis conjectured the mixing time of this process to be $O(A^2)$ which we confirm in this paper. This improves on the currently known $O(\exp (A^2))$ estimate.