The Annals of Applied Probability

Analysis of top to bottom-k shuffles

Sharad Goel

Full-text: Open access


A deck of n cards is shuffled by repeatedly moving the top card to one of the bottom kn positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as kn ranges from a constant to n. We also consider a symmetric variant of this shuffle in which at each step either the top card is randomly inserted into the bottom kn positions or a random card from the bottom kn positions is moved to the top. For this reversible shuffle we derive bounds on the L2 mixing time. Finally, we transfer mixing time estimates for the above shuffles to the lazy top to bottom-k walks that move with probability 1/2 at each step.

Article information

Ann. Appl. Probab., Volume 16, Number 1 (2006), 30-55.

First available in Project Euclid: 6 March 2006

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60
Secondary: 68

Finite Markov chains mixing time card shuffling Rudvalis shuffle


Goel, Sharad. Analysis of top to bottom- k shuffles. Ann. Appl. Probab. 16 (2006), no. 1, 30--55. doi:10.1214/10505160500000062.

Export citation


  • Aldous, D. (1983). Random walks on finite groups and rapidly mixing Markov chains. Séminaire de Probabilités XVII. Lecture Notes in Math. 986 243–297. Springer, Berlin.
  • Diaconis, P. (1988). Group Representations in Probability and Statistics. IMS, Hayward, CA.
  • Diaconis, P., Fill, J. and Pitman, J. (1992). Analysis of top to random shuffles. Combin. Probab. Comput. 1 135–155.
  • Diaconis, P. and Saloff-Coste, L. (1993). Comparison techniques for random walk on finite groups. Ann. Probab. 21 2131–2156.
  • Diaconis, P. and Saloff-Coste, L. (1993). Comparison theorems for reversible Markov chains. Ann. Appl. Probab. 3 696–730.
  • Diaconis, P. and Saloff-Coste, L. (1995). Random walks on finite groups: A survey of analytic techniques. In Probability Measures on Groups and Related Structures 11 (Z. H. Heyer, ed.) 44–75. World Scientific, River Edge, NJ.
  • Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transposition. Z. Wahrsch. Verw. Gebiete 57 159–179.
  • Diaconis, P. and Stroock, D. (1991). Geometric bounds for eigenvalues for Markov chains. Ann. Appl. Probab. 1 36–61.
  • Hildebrand, M. (1990). Rates of convergence of some random processes on finite groups. Ph.D. thesis, Harvard Univ.
  • Jonasson, J. (2005). Biased random-to-top shuffling. Preprint.
  • Saloff-Coste, L. (1996). Lectures on finite Markov chains. Lectures on Probability Theory and Statistics. Ecole d'Eté de Probabiltés de Saint-Flour XXVI. Lecture Notes in Math. 1665 301–413. Springer, Berlin.
  • Saloff-Coste, L. (2004). Random walks on finite groups. In Probability on Discrete Structures (H. Kesten, ed.). Encyclopaedia of Mathematical Sciences 110 263–346. Springer, Berlin.
  • Saloff-Coste, L. (2004). Total variation lower bounds for finite Markov chains: Wilson's lemma. In Random Walks and Geometry 515–532. de Gruyter, Berlin.
  • Wilmer, E. (2003). A local limit theorem for a family of non-reversible Markov chains. J. Theoret. Probab. 16 751–770.
  • Wilson, D. B. (2003). Mixing time of the Rudvalis shuffle. Electron. Comm. Probab. 8 77–85.
  • Wilson, D. B. (2004). Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 274–325.