Systematic versus random scans (and Hecke algebras)

Given a task, is it better to be systematic or random? I will address this ill-posed question by studying the Gibbs sampler (Glauber dynamics). Should one systematically go through the coordinates in some order or just update at random? Surprisingly, in classes of examples where things can be proved, there is little difference between these two approaches. I have studied this using group theorists' Hecke algebras and I will report much sharpened results of Bufetev, Sallez and Zhang.