623 research outputs found
Hopf algebras and Markov chains: Two examples and a theory
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.Comment: 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.
Card shuffling and diophantine approximation
The ``overlapping-cycles shuffle'' mixes a deck of cards by moving either
the th card or the th card to the top of the deck, with probability
half each. We determine the spectral gap for the location of a single card,
which, as a function of and , has surprising behavior. For example,
suppose is the closest integer to for a fixed real
. Then for rational the spectral gap is
, while for poorly approximable irrational numbers ,
such as the reciprocal of the golden ratio, the spectral gap is
.Comment: Published in at http://dx.doi.org/10.1214/07-AAP484 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
A Card Shuffling Analysis of Deformations of the Plancherel Measure of the Symmetric Group
We study deformations of the Plancherel measure of the symmetric group by
lifting them to the symmetric group and using combinatorics of card shuffling.
The existing methods for analyzing deformations of Plancherel measure are not
obviously applicable to the examples in this paper. The main idea of this paper
is to find and analyze a formula for the total variation distance between
iterations of riffle shuffles and iterations of "cut and then riffle shuffle".
Similar results are given for affine shuffles, which allow us to determine
their convergence rate to randomness
Mixing Time of the Rudvalis Shuffle
We extend a technique for lower-bounding the mixing time of card-shuffling
Markov chains, and use it to bound the mixing time of the Rudvalis Markov
chain, as well as two variants considered by Diaconis and Saloff-Coste. We show
that in each case Theta(n^3 log n) shuffles are required for the permutation to
randomize, which matches (up to constants) previously known upper bounds. In
contrast, for the two variants, the mixing time of an individual card is only
Theta(n^2) shuffles.Comment: 9 page
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