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 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)

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 ))

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) ???

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?

Relaxation time for single card

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

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

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

Eigenvalues for single card [Jonasson]

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

Eigenvalues for single card

Further reading