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Card shuffling and Diophantine approximation Omer Angel, Yuval Peres, David Wilson Annals of Applied Probability, to appear

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Overlapping cycles shuffle Deck of n cards Flip a coin to pick either n th card (bottom card) or (n-k) th card, move it to top of deck In permutation cycle notation: apply one of the following two permutations, probability ½ each: (1,2,3,4,…,n) (1,2,3,4,…,n-k)(n-k+1)…(n)

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Overlapping cycles shuffle k=1 Pick bottom card or second from bottom card, move it to the top Called Rudvalis shuffle Takes O(n 3 log n) time to mix [Hildebrand] [Diaconis & Saloff-Coste] Takes (n 3 log n) time to mix [Wilson] (with constant 1/(8 2 ))

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Generalization of Rudvalis shuffle Pick any of k bottom cards, move to top (n 3 /k 2 log n) mixing time [Goel, Jonasson] Pick either bottom card, or k th card from bottom, move to top (overlapping cycles shuffle) [Jonasson] (n 3 /k 2 log n) mixing time, no matching upper bound For k=n/2, (n 2 ) mixing time For typical k, (n log n) ???

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Mixing time of overlapping cycles shuffle Mixing time of shuffle is hard to compute, dont know the answer (open problem) Settle for modest goal of understanding the mixing of a single card Perhaps mixing time of whole permutation is O(log n) times bigger?

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Relaxation time for single card

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Markov chain for single card X t = position of card at time t By time T, card was at n-k about T/n times card was >n-k about T k/n times

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Relaxation time of card n=200 n=1000 Spikes at simple rationals

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Relaxation time for simple rational k/n

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Spectral gap for large n as k varies

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Bells have width n 3/4 : Spectral gap when k/n near simple rational

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Spectral gap and bell ensemble Thm. Relaxation time is max of all possible bells

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Eigenvalues for single card [Jonasson]

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Eigenvalues of single card in overlapping cycles shuffle n=50 k=20

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Eigenvalues for single card

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Further reading

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