Geodesic
Hopf algebras and Markov chains: two examples and a theory
Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 527-585.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explicitly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.
Classification : 05E15
Keywords: Hopf algebras, free Lie algebras, rock breaking models, shuffling 
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     title = {Hopf algebras and {Markov} chains: two examples and a theory},
     journal = {Journal of Algebraic Combinatorics},
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Diaconis, Persi; Pang, C.Y.Amy; Ram, Arun. Hopf algebras and Markov chains: two examples and a theory. Journal of Algebraic Combinatorics, Tome 39 (2014) no. 3, pp. 527-585. http://geodesic.mathdoc.fr/item/JAC_2014__39_3_a9/