The Annals of Applied Probability

Mixing time of the card-cyclic-to-random shuffle

Ben Morris, Weiyang Ning, and Yuval Peres

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Abstract

The card-cyclic-to-random shuffle on $n$ cards is defined as follows: at time $t$ remove the card with label $t$ mod $n$ and randomly reinsert it back into the deck. Pinsky [Probabilistic and combinatorial aspects of the card-cyclic-to-random shuffle (2011). Unpublished manuscript] introduced this shuffle and asked how many steps are needed to mix the deck. He showed $n$ steps do not suffice. Here we show that the mixing time is on the order of $\Theta(n\log n)$.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 5 (2014), 1835-1849.

Dates
First available in Project Euclid: 26 June 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1403812363

Digital Object Identifier
doi:10.1214/13-AAP964

Mathematical Reviews number (MathSciNet)
MR3226165

Zentralblatt MATH identifier
1321.60143

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Markov chain mixing time

Citation

Morris, Ben; Ning, Weiyang; Peres, Yuval. Mixing time of the card-cyclic-to-random shuffle. Ann. Appl. Probab. 24 (2014), no. 5, 1835--1849. doi:10.1214/13-AAP964. https://projecteuclid.org/euclid.aoap/1403812363


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References

  • [1] Bubley, R. and Dyer, M. (1997). Path coupling: A technique for proving rapid mixing in Markov Chains. In Proceedings of the 38th Annual Symposium on Foundation of Computer Science 223–231. IEEE Computer Society, Washington, DC.
  • [2] Diaconis, P. and Ram, A. (2000). Analysis of systematic scan Metropolis algorithms using Iwahori–Hecke algebra techniques. Michigan Math. J. 48 157–190.
  • [3] Diaconis, P. and Saloff-Coste, L. (1993). Comparison techniques for random walk on finite groups. Ann. Probab. 21 2131–2156.
  • [4] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
  • [5] Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables. J. Amer. Statist. Assoc. 58 13–30.
  • [6] Levin, D. A., Peres, Y. and Wilmer, E. L. (2009). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI.
  • [7] Mironov, I. (2002). (Not so) random shuffles of RC4. In Advances in Cryptology—CRYPTO 2002. 304–319. Springer, Berlin.
  • [8] Mossel, E., Peres, Y. and Sinclair, A. (2004). Shuffling by semi-random transpositions. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS’04) October 1719, 2004, Rome, Italy 572–581. IEEE Computer Society, Washington, DC.
  • [9] Pinsky, R. (2013). Probabilistic and combinatorial aspects of the card-cyclic to insertion random shuffle. Random Structures Algorithms. DOI:10.1002/rsa.20505.
  • [10] Saloff-Coste, L. and Zúñiga, J. (2007). Convergence of some time inhomogeneous Markov chains via spectral techniques. Stochastic Process. Appl. 117 961–979.
  • [11] Subag, E. (2013). A lower bound for the mixing time of the random-to-random insertions shuffle. Electron. J. Probab. 18 no. 20, 20.
  • [12] Uyemura-Reyes, J. (2002). Random walk, semi-direct products, and card shuffling. Ph.D. thesis, Stanford Univ.