The Annals of Applied Probability

A rule of thumb for riffle shuffling

Sami Assaf, Persi Diaconis, and K. Soundararajan

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We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, for example, suits disregarded or only the colors of interest. For these features the number of shuffles drops from (3/2) log2n to log2n. We derive closed formulae and an asymptotic “rule of thumb” formula which is remarkably accurate.

Article information

Ann. Appl. Probab., Volume 21, Number 3 (2011), 843-875.

First available in Project Euclid: 2 June 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization
Secondary: 60C05: Combinatorial probability

Card shuffling cutoff phenomenon lumping of Markov chains Poisson summation


Assaf, Sami; Diaconis, Persi; Soundararajan, K. A rule of thumb for riffle shuffling. Ann. Appl. Probab. 21 (2011), no. 3, 843--875. doi:10.1214/10-AAP701.

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  • [1] Aldous, D. (1983). Random walks on finite groups and rapidly mixing Markov chains. In Seminar on Probability, XVII. Lecture Notes in Math. 986 243–297. Springer, Berlin.
  • [2] Aldous, D. and Diaconis, P. (1986). Shuffling cards and stopping times. Amer. Math. Monthly 93 333–348.
  • [3] Aldous, D. and Diaconis, P. (1987). Strong uniform times and finite random walks. Adv. in Appl. Math. 8 69–97.
  • [4] Athanasiadis, C. and Diaconis, P. (2010). Functions of hyperplane walks. Adv. in Appl. Math. 45 410–437.
  • [5] Bayer, D. and Diaconis, P. (1992). Trailing the dovetail shuffle to its lair. Ann. Appl. Probab. 2 294–313.
  • [6] Boyd, S., Diaconis, P., Parrilo, P. and Xiao, L. (2005). Symmetry analysis of reversible Markov chains. Internet Math. 2 31–71.
  • [7] Ceccherini-Silberstein, T., Scarabotti, F. and Tolli, F. (2008). Harmonic Analysis on Finite Groups: Representation Theory, Gelfand Pairs and Markov Chains. Cambridge Studies in Advanced Mathematics 108. Cambridge Univ. Press, Cambridge.
  • [8] Chen, G.-Y. and Saloff-Coste, L. (2008). The cutoff phenomenon for randomized riffle shuffles. Random Structures Algorithms 32 346–374.
  • [9] Chen, G.-Y. and Saloff-Coste, L. (2010). The L2 cutoff for reversible Markov processes. J. Funct. Anal. 258 2246–2315.
  • [10] Ciucu, M. (1998). No-feedback card guessing for dovetail shuffles. Ann. Appl. Probab. 8 1251–1269.
  • [11] Conger, M. and Viswanath, D. (2006). Riffle shuffles of decks with repeated cards. Ann. Probab. 34 804–819.
  • [12] Conger, M. and Viswanath, D. (2006). Shuffling cards for blackjack, bridge, and other card games. Preprint.
  • [13] Conger, M. and Viswanath, D. (2007). Normal approximations for descents and inversions of permutations of multisets. J. Theoret. Probab. 20 309–325.
  • [14] Conger, M. A. (2007). Shuffling decks with repeated card values. Ph.D. thesis, Univ. Michigan.
  • [15] Diaconis, P. (1988). Group Representations in Probability and Statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series 11. IMS, Hayward, CA.
  • [16] Diaconis, P. (2003). Mathematical developments from the analysis of riffle shuffling. In Groups, Combinatorics and Geometry (Durham, 2001) 73–97. World Scientific, River Edge, NJ.
  • [17] Diaconis, P. and Fulman, J. (2009). Carries, shuffling, and an amazing matrix. Amer. Math. Monthly 116 788–803.
  • [18] Diaconis, P. and Holmes, S. P. (2002). Random walks on trees and matchings. Electron. J. Probab. 7 17 pp. (electronic).
  • [19] Diaconis, P., McGrath, M. and Pitman, J. (1995). Riffle shuffles, cycles, and descents. Combinatorica 15 11–29.
  • [20] Diaconis, P. and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57 159–179.
  • [21] Ding, J., Lubetzky, E. and Peres, Y. (2010). Total variation cutoff in birth and death chains. Probab. Theory Related Fields 146 61–85.
  • [22] Fässler, A. and Stiefel, E. (1992). Group Theoretical Methods and Their Applications. Birkhäuser, Boston, MA. Translated from the German by Baoswan Dzung Wong.
  • [23] Fulman, J. (2002). Applications of symmetric functions to cycle and increasing subsequence structure after shuffles. J. Algebraic Combin. 16 165–194.
  • [24] Fulman, J. (2004). A card shuffling analysis of deformations of the Plancherel measure of the symmetric group. Electron. J. Combin. 11 Research Paper 21, 15.
  • [25] Gardner, M. (1966). Martin Gardner’s New Mathematical Diversions from Scientific American. Simon and Schuster, New York.
  • [26] Gibbs, A. and Su, F. (2002). On choosing and bounding probability metrics. Int. Statis. Inst. Rev. 70 419–435.
  • [27] Gilbert, E. (1955). Theory of shuffling. Technical memorandum, Bell Laboratories, Murray Hill, NJ.
  • [28] Holte, J. M. (1997). Carries, combinatorics, and an amazing matrix. Amer. Math. Monthly 104 138–149.
  • [29] Lalley, S. (1999). Riffle shuffles and their associated dynamical systems. J. Theoret. Probab. 12 903–932.
  • [30] Reeds, J. (1976). Theory of shuffling. Unpublished manuscript, Univ. California, Berkeley.
  • [31] Reyes, J.-C. U. (2002). Random walk, semi-direct products, and card shuffling. Ph.D. thesis, Stanford Univ.
  • [32] Serre, J.-P. (1977). Linear Representations of Finite Groups. Graduate Texts in Mathematics 42. Springer, New York. Translated from the second French edition by Leonard L. Scott.
  • [33] Weaver, J. R. (1985). Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, and eigenvectors. Amer. Math. Monthly 92 711–717.