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 L X L region we bound the mixing time by O(L^4 log L), which improves on the previous bound of O(L^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.
We propose a model of card shuffling where a pack of cards, spread as points on a square table, are repeatedly gathered locally at random spots and then spread towards a random direction. A shuffling of the cards is then obtained by arranging the cards by their increasing $x$-coordinate values. When there are $m$ cards on the table we show that this random ordering gets mixed in time $O\left(\log m\right)$. Explicit constants are evaluated in a diffusion limit when the position of $m$ cards evolves as an interesting $2m$-dimensional non-reversible reflected jump diffusion in time. Our main tec…
We consider the biased card shuffling and the Asymmetric Simple Exclusion Process (ASEP) on the segment. We obtain the asymptotic of their mixing times: our result show that these two continuous-time Markov chains display cutoff. Our analysis combines several ingredients including: a study of the hydrodynamic profile for ASEP, the use of monotonic eigenfunctions, stochastic comparisons and concentration inequalities.
The card-cyclic-to-random shuffle is the card shuffle where the $n$ cards are labeled $1,\ldots,n$ according to their starting positions. Then the cards are mixed by first picking card $1$ from the deck and reinserting it at a uniformly random position, then repeating for card $2$, then for card $3$ and so on until all cards have been reinserted in this way. Then the procedure starts over again, by first picking the card with label $1$ and reinserting, and so on. Morris, Ning and Peres \cite{MNP} recently showed that the order of the number of shuffles needed to mix the deck in this way is $n\…
The ``overlapping-cycles shuffle'' mixes a deck of $n$ cards by moving either the $n$th 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 $α\in(0,1)$. 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 $Θ(n^{-3/2})$.
In this expository article, we highlight the direct connection between card shuffling and the functions known as $P$-partitions that come from algebraic combinatorics. While many (but not all) of the results we discuss are known, we give a unified treatment. The key idea is this: the probability of obtaining a permutation $π$ from shelf shuffling is the probability that a random $P$-partition is sorted by $π$, and the probability of obtaining $π$ from riffle shuffling is the probability that a random $P$-partition is sorted by $π^{-1}$.
We study Markov chains which model genome rearrangements. These models are useful for studying the equilibrium distribution of chromosomal lengths, and are used in methods for estimating genomic distances. The primary Markov chain studied in this paper is the top-swap Markov chain. The top-swap chain is a card-shuffling process with $n$ cards divided over $k$ decks, where the cards are ordered within each deck. A transition consists of choosing a random pair of cards, and if the cards lie in different decks, we cut each deck at the chosen card and exchange the tops of the two decks. We prove p…
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.
We study deformations of the Plancherel measure of the symmetric group by lifting them to the symmetric group and using combinatorics of card shuffling. The existing methods for analyzing deformations of Plancherel measure are not obviously applicable to the examples in this paper. The main idea of this paper is to find and analyze a formula for the total variation distance between iterations of riffle shuffles and iterations of "cut and then riffle shuffle". Similar results are given for affine shuffles, which allow us to determine their convergence rate to randomness.
This paper considers the effect of riffle shuffling on decks of cards, allowing for some cards to be indistinguishable from other cards. The dual problem of dealing a game with hands, such as bridge or poker, is also considered. The Gilbert-Shannon-Reeds model of card shuffling is used, along with variation distance for measuring how close to uniform a deck has become. The surprising results are that for a deck with only two types of cards (such as red and black), the shuffler can greatly improve the randomness of the deck by insuring that the top and bottom cards are the same before shuffling…
Consider the following method of card shuffling. Start with a deck of $N$ cards numbered 1 through N. Fix a parameter $p$ 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. 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 not equal to 1/2, the mixing time of this card shuffling is O…
In card games, in casino games with multiple decks of cards and in cryptography, one is sometimes faced with the following problem: how can a human (as opposed to a computer) shuffle a large deck of cards? The procedure we study is to break the deck into several reasonably sized piles, shuffle each thoroughly, recombine the piles, do some simple deterministic operation, for instance a cut, and repeat. This process can also be seen as a generalised Bernoulli-Laplace urn model. We use coupling arguments and spherical function theory to derive upper and bounds on the mixing times of these Markov …
This paper is about the following question: How many riffle shuffles mix a deck of card for games such as blackjack and bridge? An object that comes up in answering this question is the descent polynomial associated with pairs of decks, where the decks are allowed to have repeated cards. We prove that the problem of computing the descent polynomial given a pair of decks is $#P$-complete. We also prove that the coefficients of these polynomials can be approximated using the bell curve. However, as must be expected in view of the $#P$-completeness result, approximations using the bell curve are …
Recently, Diaconis, Ram and I created Markov chains out of the coproduct-then-product operator on combinatorial Hopf algebras. These chains model the breaking and recombining of combinatorial objects. Our motivating example was the riffle-shuffling of a deck of cards, for which this Hopf algebra connection allowed explicit computation of all the eigenfunctions. The present note replaces in this construction the coproduct-then-product map with convolutions of projections to the graded subspaces, effectively allowing us to dictate the distribution of sizes of the pieces in the breaking step of t…
The number of ``carries'' when $n$ random integers are added forms a Markov chain [23]. We show that this Markov chain has the same transition matrix as the descent process when a deck of $n$ cards is repeatedly riffle shuffled. This gives new results for the statistics of carries and 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 …
We study how many riffle shuffles are required to mix n cards if only certain features of the deck are of interest, e.g. suits disregarded or only the colors of interest. For these features, the number of shuffles drops from 3/2 log_2(n) to log_2(n). We derive closed formulae and an asymptotic `rule of thumb' formula which is remarkably accurate.
In the cyclic-to-random shuffle, we are given n cards arranged in a circle. At step k, we exchange the k'th card along the circle with a uniformly chosen random card. The problem of determining the mixing time of the cyclic-to-random shuffle was raised by Aldous and Diaconis in 1986. Recently, Mironov used this shuffle as a model for the cryptographic system known as ``RC4'' and proved an upper bound of O(n log n) for the mixing time. We prove a matching lower bound, thus establishing that the mixing time is indeed of order $Θ(n \log n)$. We also prove an upper bound of O(n log n) for the mixi…
Many casinos routinely use mechanical card shuffling machines. We were asked to evaluate a new product, a shelf shuffler. This leads to new probability, new combinatorics and to some practical advice which was adopted by the manufacturer. The interplay between theory, computing, and real-world application is developed.
The "carries" when n random numbers are added base b form a Markov chain with an "amazing" transition matrix determined by Holte. This same Markov chain occurs in following the number of descents or rising sequences when n cards are repeatedly riffle shuffled. We give generating and symmetric function proofs and determine the rate of convergence of this Markov chain to stationarity. Similar results are given for type B shuffles. We also develop connections with Gaussian autoregressive processes and the Veronese mapping of commutative algebra.
E. Thorp introduced the following card shuffling model. 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 show that if $n$ is a power of 2 then the mixing time of the Thorp shuffle is $O(\log^3 n)$. Previously, the best known bound was $O(\log^4 n)$.
We extend a technique for lower-bounding the mixing time of card-shuffling Markov chains, and use it to bound the mixing time of the Rudvalis Markov chain, as well as two variants considered by Diaconis and Saloff-Coste. We show that in each case Theta(n^3 log n) shuffles are required for the permutation to randomize, which matches (up to constants) previously known upper bounds. In contrast, for the two variants, the mixing time of an individual card is only Theta(n^2) shuffles.
The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution a…
Consider a real hyperplane arrangement and let $\mathcal{C}$ denote the occurring chambers. Bidigare, Hanlon and Rockmore introduced a Markov chain on $\mathcal{C}$ which is a generalization of some card shuffling models used in computer science, biology and card games. This paper introduces strong stationary arguments for this Markov chain, which provide explicit bounds for the separation distance.
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(log^4 n). Previously, the best …
Card shuffling models have provided simple motivating examples for the mathematical theory of mixing times for Markov chains. As a complement, we introduce a more intricate realistic model of a certain observable real-world scheme for mixing human players onto teams. We quantify numerically the effectiveness of this mixing scheme over the 7 or 8 steps performed in practice. We give a combinatorial proof of the non-trivial fact that the chain is indeed irreducible.
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 $n^2$ and at most of order $n^2\log n$. 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 $n^2\log n$, thereby showing that $n^2\log n$ is indeed the correct order of the mixing time. It is our …
A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card shuffling (Aldous-Diaconis, 1986), this phenomenon is now believed to be rather typical among fast mixing Markov chains. Yet, establishing it rigorously often requires a challengingly detailed understanding of the underlying chain. Here we consider non-backtracking random walks on random graphs with a given degree sequence. Under a general sparsity conditio…
Discovered in the context of card shuffling by Aldous, Diaconis and Shahshahani, the cutoff phenomenon has since then been established in a variety of Markov chains. However, proving cutoff remains a delicate affair, which requires a detailed knowledge of the chain. Identifying the general mechanisms underlying this phase transition -- without having to pinpoint its precise location -- remains one of the most fundamental open problems in the area of mixing times. In the present paper, we make a step in this direction by establishing cutoff for Markov chains with non-negative curvature, under a…
This thesis introduces a way to build Markov chains out of Hopf algebras. The transition matrix of a "Hopf-power Markov chain" is (the transpose of) the matrix of the coproduct-then-product operator on a combinatorial Hopf algebra with respect to a suitable basis. These chains describe the breaking-then-recombining of the combinatorial objects in the Hopf algebra. The motivating example is the famous Gilbert-Shannon-Reeds model of riffle-shuffling of a deck of cards, which arises in this manner from the shuffle algebra. The primary reason for constructing Hopf-power Markov chains, or for rephr…
