Card shuffling and Diophantine approximation

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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-k1)(n)

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Overlapping cycles shuffle k1

• 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 kn/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, 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 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 gtn-k about T k/n times
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Relaxation time of card
n1000
n200
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 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
n50 k20
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Eigenvalues for single card
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Further reading

• http//arxiv.org/abs/0707.2994