Card Shuffling Research Articles

A Better Way to Deal the Cards

Most of the work on card shuffling assumes that all the cards in a deck are distinct, and that in a well-shuffled deck all orderings need to be equally likely. We consider the case of decks with repeated cards and decks which are dealt into hands, as in bridge and poker. We derive asymptotic formulas for the randomness of the resulting games. Results include the influence of where a poker deck is cut, and the fact that switching from cyclic dealing to back-and-forth dealing will improve the randomness of a bridge deck by a factor of 13.

Convergence time to the Ewens sampling formula in the infinite alleles Moran model

In this paper, we establish an upper bound for time to convergence to stationarity for the discrete time infinite alleles Moran model. If M is the population size and mu is the mutation rate, this bound gives a cutoff time of log(Mmu)/mu generations. The stationary distribution for this process in the case of sampling without replacement is the Ewens sampling formula. We show that the bound for the total variation distance from the generation t distribution to the Ewens sampling formula is well approximated by one of the extreme value distributions, namely, a standard Gumbel distribution. Beginning with the card shuffling examples of Aldous and Diaconis and extending the ideas of Donnelly and Rodrigues for the two allele model, this model adds to the list of Markov chains that show evidence for the cutoff phenomenon. Because of the broad use of infinite alleles models, this cutoff sets the time scale of applicability for statistical tests based on the Ewens sampling formula and other tests of neutrality in a number of population genetic studies.

Improved mixing time bounds for the Thorp shuffle and L-reversal chain

We prove a theorem that reduces bounding the mixing time of a card shuffle to verifying a condition that involves only pairs of cards, then we use it to obtain improved bounds for two previously studied models. E. Thorp introduced the following card shuffling model in 1973: Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We obtain a mixing time bound of O(log4n). Previously, the best known bound was O(log29n) and previous proofs were only valid for n a power of 2. We also analyze the following model, called the L-reversal chain, introduced by Durrett: There are n cards arrayed in a circle. Each step, an interval of cards of length at most L is chosen uniformly at random and its order is reversed. Durrett has conjectured that the mixing time is O(max(n, n3/L3)log n). We obtain a bound that is within a factor O(log2n) of this, the first bound within a poly log factor of the conjecture.

Card shuffling and Diophantine approximation

The “overlapping-cycles shuffle” mixes a deck of n cards by moving either the nth card or the (n−k)th card to the top of the deck, with probability half each. We determine the spectral gap for the location of a single card, which, as a function of k and n, has surprising behavior. For example, suppose k is the closest integer to αn for a fixed real α∈(0, 1). Then for rational α the spectral gap is Θ(n−2), while for poorly approximable irrational numbers α, such as the reciprocal of the golden ratio, the spectral gap is Θ(n−3/2).

Robust Mixing

In this paper, we develop a new "robust mixing" framework for reasoning about adversarially modified Markov Chains (AMMC). Let $\mathbb{P}$ be the transition matrix of an irreducible Markov Chain with stationary distribution $\pi$. An adversary announces a sequence of stochastic matrices $\{\mathbb{A}_t\}_{t > 0}$ satisfying $\pi\mathbb{A}_t = \pi$. An AMMC process involves an application of $\mathbb{P}$ followed by $\mathbb{A}_t$ at time $t$. The robust mixing time of an ergodic Markov Chain $\mathbb{P}$ is the supremum over all adversarial strategies of the mixing time of the corresponding AMMC process. Applications include estimating the mixing times for certain non-Markovian processes and for reversible liftings of Markov Chains. Non-Markovian card shuffling processes: The random-to-cyclic transposition process is a non-Markovian card shuffling process, which at time $t$, exchanges the card at position $L_t := t {\pmod n}$ with a random card. Mossel, Peres and Sinclair (2004) showed a lower bound of $(0.0345+o(1))n\log n$ for the mixing time of the random-to-cyclic transposition process. They also considered a generalization of this process where the choice of $L_t$ is adversarial, and proved an upper bound of $C n\log n + O(n)$ (with $C \approx 4\times 10^5$) on the mixing time. We reduce the constant to $1$ by showing that the random-to-top transposition chain (a Markov Chain) has robust mixing time $\leq n\log n + O(n)$ when the adversarial strategies are limited to holomorphic strategies, i.e. those strategies which preserve the symmetry of the underlying Markov Chain. We also show a $O(n\log^2 n)$ bound on the robust mixing time of the lazy random-to-top transposition chain when the adversary is not limited to holomorphic strategies. Reversible liftings: Chen, Lovasz and Pak showed that for a reversible ergodic Markov Chain $\mathbb{P}$, any reversible lifting $\mathbb{Q}$ of $\mathbb{P}$ must satisfy $\mathcal{T}(\mathbb{P}) \leq \mathcal{T}(\mathbb{Q})\log (1/\pi_*)$ where $\pi_*$ is the minimum stationary probability. Looking at a specific adversarial strategy allows us to show that $\mathcal{T}(\mathbb{Q}) \geq r(\mathbb{P})$ where $r(\mathbb{P})$ is the relaxation time of $\mathbb{P}$. This gives an alternate proof of the reversible lifting result and helps identify cases where reversible liftings cannot improve the mixing time by more than a constant factor.

Random orderings of the integers and card shuffling

In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.

Two mathematical notes – new homogenised Simpson's rules and a riffle shuffle conjecture

PurposeSeeks to derive a class of “homogeneous” rules for numerical integration from earlier results and empirical findings to treat the apparently magical reordering of a pack of cards after successive shuffles, previously discussed by Zeeberg, from a new angle and to form a mathematical conjecture.Design/methodology/approachThe studies were made using computer programs for which the language JavaScript proved adequate.FindingsThe rules for numerical integration are more precise than earlier versions. The conjecture associated with card shuffling appears to be novel.Practical implicationsImproved methods of numerical integration have practical value in many areas. The conjecture is in the field of number theory, with no obvious immediate applications.Originality/valueThe findings and methods are original. The demonstration of a plausible mathematical conjecture may provoke further studies aimed at its proof as a theorem, or its refutation.

Biased random-to-top shuffling

Recently Wilson [Ann. Appl. Probab. 14 (2004) 274–325] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilson’s technique to find lower bounds of the correct order for card shuffling Markov chains where at each time step a random card is picked and put at the top of the deck. Two classes of such shuffles are addressed, one where the probability that a given card is picked at a given time step depends on its identity, the so-called move-to-front scheme, and one where it depends on its position. For the move-to-front scheme, a test function that is a combination of several different eigenvectors of the transition matrix is used. A general method for finding and using such a test function, under a natural negative dependence condition, is introduced. It is shown that the correct order of the mixing time is given by the biased coupon collector’s problem corresponding to the move-to-front scheme at hand. For the second class, a version of Wilson’s technique for complex-valued eigenvalues/eigenvectors is used. Such variants were presented in [Random Walks and Geometry (2004) 515–532] and [Electron. Comm. Probab. 8 (2003) 77–85]. Here we present another such variant which seems to be the most natural one for this particular class of problems. To find the eigenvalues for the general case of the second class of problems is difficult, so we restrict attention to two special cases. In the first case the card that is moved to the top is picked uniformly at random from the bottom k=k(n)=o(n) cards, and we find the lower bound (n3/(4π2k(k−1)))logn. Via a coupling, an upper bound exceeding this by only a factor 4 is found. This generalizes Wilson’s [Electron. Comm. Probab. 8 (2003) 77–85] result on the Rudvalis shuffle and Goel’s [Ann. Appl. Probab. 16 (2006) 30–55] result on top-to-bottom shuffles. In the second case the card moved to the top is, with probability 1/2, the bottom card and with probability 1/2, the card at position n−k. Here the lower bound is again of order (n3/k2)logn, but in this case this does not seem to be tight unless k=O(1). What the correct order of mixing is in this case is an open question. We show that when k=n/2, it is at least Θ(n2).

The overhand shuffle mixes in Θ(n2logn) steps

The overhand shuffle is one of the “real” card shuffling methods in the sense that some people actually use it to mix a deck of cards. A mathematical model was constructed and analyzed by Pemantle [J. Theoret. Probab. 2 (1989) 37–49] who showed that the mixing time with respect to variation distance is at least of order n2 and at most of order n2logn. In this paper we use an extension of a lemma of Wilson [Ann. Appl. Probab. 14 (2004) 274–325] to establish a lower bound of order n2logn, thereby showing that n2logn is indeed the correct order of the mixing time. It is our hope that the extension of Wilson’s lemma will prove useful also in other situations; it is demonstrated how it may be used to give a simplified proof of the Θ(n3logn) lower bound of Wilson [Electron. Comm. Probab. 8 (2003) 77–85] for the Rudvalis shuffle.

Mixing times of the biased card shuffling and the asymmetric exclusion process

Consider the following method of card shuffling. Start with a deck of N N cards numbered 1 through N N . Fix a parameter p p between 0 and 1. In this model a “shuffle” consists of uniformly selecting a pair of adjacent cards and then flipping a coin that is heads with probability p p . If the coin comes up heads, then we arrange the two cards so that the lower-numbered card comes before the higher-numbered card. If the coin comes up tails, then we arrange the cards with the higher-numbered card first. In this paper we prove that for all p ≠ 1 / 2 p\ne 1/2 , the mixing time of this card shuffling is O ( N 2 ) O(N^2) , as conjectured by Diaconis and Ram (2000). Our result is a rare case of an exact estimate for the convergence rate of the Metropolis algorithm. A novel feature of our proof is that the analysis of an infinite (asymmetric exclusion) process plays an essential role in bounding the mixing time of a finite process.

Card shuffling and the decomposition of tensor products

Let H be a subgroup of a .nite group G. We use Markov chains to quantify how large r should be so that the decomposition of the r tensor power of the representation of G on cosets on H behaves (after renormalization) like the regular representation of G. For the case where G is a symmetric group and H a parabolic subgroup, we .nd that this question is precisely equivalent to the question of how large r should be so that r iterations of a shu.ing method randomise the Robinson-Schensted-Knuth shape of a permutation. This equivalence is remarkable, if only because the representation theory problem is related to a reversible Markov chain on the set of representations of the symmetric group, whereas the card shu.ing problem is related to a nonreversible Markov chain on the symmetric group. The equivalence is also useful, and results on card shu.ing can be applied to yield sharp results about the decomposition of tensor powers.

A Card Shuffling Analysis of Deformations of the Plancherel Measure of the Symmetric Group

This paper finds and analyzes a formula for the total variation distance between iterations of riffle shuffles and iterations of "cut and then riffle shuffle". This allows one to obtain information about the convergence rate of permutation statistics (such as the length of the longest increasing subsequence or position of a given card) under these processes. Similar results are given for affine shuffles, which allow us to determine their convergence rate to randomness.

Mixing times of lozenge tiling and card shuffling Markov chains

We show how to combine Fourier analysis with coupling arguments to bound the mixing times of a variety of Markov chains. The mixing time is the number of steps a Markov chain takes to approach its equilibrium distribution. One application is to a class of Markov chains introduced by Luby, Randall and Sinclair to generate random tilings of regions by lozenges. For an $\ell\times\ell$ region we bound the mixing time by $O(\ell^4\log\ell)$, which improves on the previous bound of $O(\ell^7)$, and we show the new bound to be essentially tight. In another application we resolve a few questions raised by Diaconis and Saloff-Coste by lower bounding the mixing time of various card-shuffling Markov chains. Our lower bounds are within a constant factor of their upper bounds. When we use our methods to modify a path-coupling analysis of Bubley and Dyer, we obtain an $O(n^3\log n)$ upper bound on the mixing time of the Karzanov--Khachiyan Markov chain for linear extensions.

Applications of the Brauer Complex: Card Shuffling, Permutation Statistics, and Dynamical Systems

By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type A are found. This leads to new models of card shuffling and has interesting combinatorial and number-theoretic consequences. An analysis of type C gives another solution to a problem of Rogers in dynamical systems: the enumeration of unimodal permutations by cycle structure. The proof uses the factorization theory of palindromic polynomials over finite fields. Contact is made with symmetric function theory.

On transience of card shuffling

We present simple proofs of transience/recurrence for certain card shuffling models, that is, random walks on the infinite symmetric group. 1. CARD SHUFFLING MODELS In this note, we consider several models of shuffling an infinite deck of cards. One of these models has been considered previously by Lawler [La]; our methods (using flows, shorting and comparison of Dirichlet forms) will partially simplify his result. Card shuffling is formalized by performing successive i.i.d. random permutations in the group S00 of all permutations of the positive integers N that leave all but finitely many elements fixed. We identify the symmetric group Sn with the subgroup of S00 fixing all elements > n, so that S00 is the union of the S, We read the product of two permutations x, y from left to right, that is, x y sends j E N to y(x(j)). Let ,u be a symmetric probability measure on S00 whose support generates the whole group. Associate with it a sequence Xn, n > 1, of i.i.d. S00-valued random variables with common distribution ,u, and consider the associated random walk Zn = Xl ... Xn. This means that we start with the deck of cards in order (Zo is the identity), and at each step we choose a random permutation Xn according to ,u which tells us how to shuffle once more what we had obtained previously. The question addressed here is the following. Will the deck of cards eventually return to its original order with probability one (recurrence), or is this probability strictly smaller than 1 (transience)? We now describe four shuffling models, that is, probabilities t1i, ... W, 4, each one governing another random walk. We start by considering a probability distribution p ) on {2, 3, ...}. 1) First choose n with probability p(n) and then j E {1, ... , n 1} with probability 1/(n 1), and exchange the n-th with the j-th card. Writing t(n,j) for the transposition of n and j, we have o0 n-1 A = , 4, (n) at(nj) ri=2 j=1 Received by the editors March 12, 1999 and, in revised form, July 21, 1999. 2000 Mathematics Subject Classification. Primary 60G50, 60J10; Secondary 60B15.

Semisimple Orbits of Lie Algebras and Card-Shuffling Measures on Coxeter Groups

Random walk on the chambers of hyperplane arrangements is used to define a family of card shuffling measures HW,x for a finite Coxeter group W and real x≠0. By algebraic group theory, there is a map Φ from the semisimple orbits of the adjoint action of a finite group of Lie type on its Lie algebra to the conjugacy classes of the Weyl group. Choosing such a semisimple orbit uniformly at random thereby induces a probability measure on the conjugacy classes of the Weyl group. For types A, B, and the identity conjugacy class of W for all types, it is proved that for q very good, this measure on conjugacy classes is equal to the measure arising from HW,q. The possibility of refining Φ to a map to elements of the Weyl group is discussed.

Counting Semisimple Orbits of Finite Lie Algebras by Genus

The adjoint action of a finite group of Lie type on its Lie algebra is studied. A simple formula is conjectured for the number of semisimple orbits of a given split genus. This conjecture is proved for type A, and partial results are obtained for other types. For type A a probabilistic interpretation is given in terms of card shuffling.

Magic tricks, card shuffling and dynamic computer memories

Introduction 1. The perfect shuffle 2. The order of shuffles 3. Shuffle groups 4. Generalizing the perfect shuffle 5. Dynamic computer memories. Appendix 1. The order of shuffles Appendix 2. How to do the Faro shuffle Appendix 3. Tours on decks of size 8, 16, 32 and 64 Appendix 4. A Lagniappe Bibliography.

The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

Convergence Rates for Markov Chains

This is an expository paper that presents various ideas related to nonasymptotic rates of convergence for Markov chains. Such rates are of great importance for stochastic algorithms that are widely used in statistics and in computer science. They also have applications to analysis of card shuffling and other areas.In this paper, we attempt to describe various mathematical techniques that have been used to bound such rates of convergence. In particular, we describe eigenvalue analysis, random walks on groups, coupling, and minorization conditions. Connections are made to modern areas of research wherever possible. Elements of linear algebra, probability theory, group theory, and measure theory are used, but efforts are made to keep the presentation elementary and accessible.