Abstract
The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut `about in half' and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function. Le modèle de Gilbert-Shannon-Reeds pour mélange de cartes suppose que les cartes sont d'abord coupées environ de moitié, puis intercalées ensemble. Nous analysons une variante naturelle, où la coupe initiale est biaisée. En proposant une une extension des résultats de Fulman (1998), nous montrons une forte coupure dans les distances de séparation et à l'infinité L. Cette analyse est possible grâce à l'étroite relation entre brassage et fonctions quasi-symétriques.
Highlights
We analyze a natural one-parameter model for riffle shuffling a deck of n cards
The deck is cut into two piles with a binomial (n, θ) distribution
The piles are riffled together sequentially according to the following rule: if the left pile has A cards and the right pile has B cards, drop the card from the bottom of the left pile with probability A/(A + B)
Summary
We analyze a natural one-parameter model for riffle shuffling a deck of n cards. Both SEP(k) and ∞(k) are upper bounds for the total variation metric:. Theorem 1 For the θ-biased riffle shuffle measure on Sn, let 2 log n − log 2 + c k = − log(θ2 + (1 − θ)). An upper bound on separation of this form is given in [Fulman(1998)]. There is an intimate connection between these biased shuffles and quasisymmetric functions explained in Section 3 where we prove (1.4) and (1.5). The upper bound in [Fulman(1998)] is derived using a strong stationary time.
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