card shuffling and diophantine approximation omer angel, yuval peres, david wilson annals of applied...

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