Download presentation
Presentation is loading. Please wait.
Published byJaquelin Sidley Modified over 10 years ago
1
Card shuffling and Diophantine approximation Omer Angel, Yuval Peres, David Wilson Annals of Applied Probability, to appear
2
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)
3
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 ))
4
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) ???
5
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?
6
Relaxation time for single card
7
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
10
Relaxation time of card n=200 n=1000 Spikes at simple rationals
11
Relaxation time for simple rational k/n
12
Spectral gap for large n as k varies
13
Bells have width n 3/4 : Spectral gap when k/n near simple rational
14
Spectral gap and bell ensemble Thm. Relaxation time is max of all possible bells
15
Eigenvalues for single card [Jonasson]
16
Eigenvalues of single card in overlapping cycles shuffle n=50 k=20
17
Eigenvalues for single card
19
Further reading http://arxiv.org/abs/0707.2994
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.