A Markov Chain Arising from the Hopf Square Map on a Non-cocommutative Quantum Group

Expanding upon the rich history of algebraic techniques in probability, we show the existence of and construct a Markov chain using the Hopf square map on a quantum group that is both non-commutative and non-cocommutative. This extends the work of Diaconis, Pang, and Ram to other Hopf algebras. The new, one-dimensional chain requires different analytical approaches. In this case we use standard martingale theory to prove the existence of a phase transition and prove bounds on the expected growth rates.