Imagine n particles labelled 1 to n on a ring with n sites. A particle i can swap place with its left neighbor j if i<j. This becomes a Markov chain model which has been studied by many authors and I will explain some very nice probabilistic and combinatorial properties, e.g. by Arvind Ayyer and myself.
I will also report on more recent work, joint with Erik Aas, on a continuous version of this model.