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The operation of squaring (coproduct followed by product) in a combinatorialHopf algebra is shown to induce a Markov chain in natural bases. Chainsconstructed in this way include widely studied methods of card shuffling, anatural "rock-breaking" process, and Markov chains on simplicial complexes.Many of these chains can be explictly diagonalized using the primitive elementsof the algebra and the combinatorics of the free Lie algebra. For cardshuffling, this gives an explicit description of the eigenvectors. Forrock-breaking, an explicit description of the quasi-stationary distribution andsharp rates to absorption follow.

Hopf algebras and Markov chains: Two examples and a theory