Card shuffling and Diophantine approximation Omer Angel, Yuval Peres, David Wilson Annals of Applied...
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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 nth 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(n3 log n) time to mix [Hildebrand]
[Diaconis & Saloff-Coste]• Takes (n3 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 (n3/k2 log n) mixing time [Goel, Jonasson]
• Pick either bottom card, or kth card from bottom, move to top (overlapping cycles shuffle) [Jonasson]
(n3/k2 log n) mixing time, no matching upper bound
For k=n/2, (n2) mixing time For typical k, (n log n) ???
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Mixing time ofoverlapping cycles shuffle
• Mixing time of shuffle is hard to compute, don’t 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 timefor single card
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Markov chain for single card• Xt = 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 n3/4: Spectral gap when k/n “near” simple rational
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Spectral gap and bell ensembleThm. 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=50k=20
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Eigenvalues for single card
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Further reading
http://arxiv.org/abs/0707.2994