A Gibbs sampler on the n-simplex

We determine the mixing time of a simple Gibbs sampler on the unit simplex, confirming a conjecture of Aldous. The upper bound is based on a two-step coupling, where the first step is a simple contraction argument and the second step is a non-Markovian coupling. We also present a MCMC-based perfect sampling algorithm based on our proof which can be applied with Gibbs samplers that are harder to analyze.