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Information Center for Mathematical Science

PAC

Information Center for Mathematical Science

PAC

Hopf algebras and Markov chains: Two examples and a theory
Author Persi Diaconis (Stanford University)
Homepage Url http://www-stat.stanford.edu/~cgates/PERSI/year.html
Coauthors Amy Pang, Arun Ram
Abstract 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 shuing, a natural \rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuing, 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.
Abstract Url http://arxiv.org/pdf/1206.3620v1.pdf