Card shuffling and Diophantine approximation

Omer Angel, Yuval Peres, David Wilson

Annals of Applied Probability, to appear

“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)

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))

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) ???

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?

Relaxation timefor single card

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

Relaxation time of card

n=200 n=1000

Spikes at “simple” rationals

Relaxation time for simple rational k/n

Spectral gap for large n as k varies

Bells have width n3/4: Spectral gap when k/n “near” simple rational

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

Eigenvalues for single card

[Jonasson]

Eigenvalues of single card in overlapping cycles shuffle

n=50k=20

Eigenvalues for single card

Further reading

http://arxiv.org/abs/0707.2994