Download presentation

Presentation is loading. Please wait.

Published byJaquelin Sidley Modified over 3 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

OK

Equation Jeopardy Add Mixed Multiply/ Divide Fractions Q $100 Q $200 Q $300 Q $400 Q $500 Q $100 Q $200 Q $300 Q $400 Q $500 Final Jeopardy Subtract.

Equation Jeopardy Add Mixed Multiply/ Divide Fractions Q $100 Q $200 Q $300 Q $400 Q $500 Q $100 Q $200 Q $300 Q $400 Q $500 Final Jeopardy Subtract.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on nature and scope of human resource management Ppt on supply chain management of nokia phones Ppt on high power transmission Ppt on polynomials of 99 Appt only ph clinics Isometric view ppt online Ppt on history of olympics torch Ppt on air pollution for class 6 Ppt on statistics for class 11th Download ppt on query processing and optimization