Hopf algebras and Markov chains: Two examples and a theory
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 shuffling, 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 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.
Keywords
Cite
@article{arxiv.1206.3620,
title = {Hopf algebras and Markov chains: Two examples and a theory},
author = {Persi Diaconis and C. Y. Amy Pang and Arun Ram},
journal= {arXiv preprint arXiv:1206.3620},
year = {2015}
}
Comments
51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.)