Saul Jacka, Jon Warren
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.
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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 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.
A Random Card Shuffling Process
该研究针对含n红n黑共2n张牌的随机牌组，分析通过特定移牌规则得到颜色交替牌组的平均操作次数，借助多领域数学工具将过程建模为有限马尔可夫链。
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…
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.
On the position matrix of single-shelf shuffle and card guessing
本文研究了单货架洗牌的位置矩阵的谱性质，确定了其完整谱和特征空间，并给出了无反馈猜牌游戏的最优猜测策略。
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.
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\…
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…
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 $Θ(n^{-3/2})$.
A central limit theorem for a card shuffling problem
本文研究随机排列中连续整数块反复合并至只剩一个元素所需的排列次数，给出其期望、方差及所有更高中心矩的显式渐近表达式，并证明其满足中心极限定理。
Optimal complete-feedback card guessing under nonincreasing valley weights: A proof of the Diaconis-Fulman-Holmes shelf-shuffling conjecture
该论文证明了Diaconis-Fulman-Holmes关于货架洗牌后方向跟踪策略最优的猜想，通过更一般的非递增谷权重定理，利用保持前缀的相邻交换，最大化条件概率与期望正确猜测数。
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…
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.
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 Sal…
Cutoff profile of the Metropolis biased card shuffling
本文证明了 Metropolis 偏置洗牌（多物种 ASEP）的截断窗口为 N^{1/3} 量级，截断轮廓由 GOE Tracy-Widom 分布给出，证实了 Bufetov-Nejjar 猜想，并推广了有向交换过程的收敛结果。
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.
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…
The $k$-cycle shuffling with repeated cards
本文针对含重复牌的$k$-循环洗牌，通过构造易处理的辅助测度推广已有结果，分别在固定$l$和$l$慢增长情形下证明了全变差混合的截止现象并给出对应极限轮廓。
Shuffling via sums of Jucys--Murphy Elements
本文研究了一种通过对称群S_m的Jucys-Murphy元素对应的置换进行洗牌的家族，确定了这些洗牌的转移矩阵的本征值，并研究了特定情况下的混合时间，证明了k-星置换洗牌在总变差截止时具有特定的渐进行为。
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.
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 …
The random-to-top and the riffle shuffle are two well-studied methods for shuffling a deck of cards. These correspond to the symmetric group $S_n$, i.e., the Coxeter group of type $A_{n-1}$. In this paper, we give analogous shuffles for the Coxeter groups of type $B_n$ and $D_n$. These can be interpreted as shuffles on a ``signed'' deck of cards. With these examples as motivation, we abstract the notion of a shuffle algebra which captures the connection between the algebraic structure of the shuffles and the geometry of the Coxeter groups. We also briefly discuss the generalisation to building…
Limit theorems for descents and inversions of shelf-shuffles
针对货架洗牌产生的排列，证明了下降次数和逆序数的中心极限定理，并给出了逆序收敛速度的与货架数无关的界。
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 $O(\log^4 n)$.
Confidentiality in a Card-Based Protocol Under Repeated Biased Shuffles
本文针对五张牌戏法协议，通过条件概率和马尔可夫链特征结构分析有偏随机切牌导致的信息泄露，并推导出将泄露降至可接受水平所需重复切牌次数的紧界。
Universality of Cutoff for Riffle Shuffling
本文证明Riffle洗牌在一般切牌分布下混合时间具有截断现象，混合时间与$\log N$之比收敛到显式常数，并推广到多部分洗牌，揭示了冷点现象的普适性。
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.
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 …
Hitting time mixing for the random transposition walk
该研究针对随机对换洗牌的混合时间问题，证实了N. Berestycki的猜想，证明所有牌首次被全部触碰的击中时刻下牌堆分布接近均匀，且该时刻是最优的。
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.
Consider n cards that are labeled 1 through n with n an even integer. The cards are put face down and their ordering starts with card labeled 1 on top through card labeled n at the bottom. The cards are top to random shuffled m times and placed face down on the table. Starting from the top the cards are guessed without feedback (i.e. whether the guess was correct or false and what the guessed card was) one at a time. For m > 4nlog n+cn we find a guessing strategy that maximizes the expected number of correct guesses.
Cutoff in total variation for the shelf shuffle
本文证明搁架洗牌器混合一副n张牌在全变差下需要且仅需(5/4)log_{2m}n次洗牌，并确定了常数窗口的截止现象及其剖面。
Mixing time of the torus shuffle
本文证明了一个将洗牌混合时间简化为三张牌条件验证的定理，并应用于Diaconis提出的环面洗牌模型，得到与猜想$O(n^3\log n)$仅差多对数因子的界。
The one-sided cycle shuffles in the symmetric group algebra
研究对称群代数中的单侧洗牌算子，计算其特征值与重数，证明可对角化条件，并给出随机到下方洗牌的强平稳时间。
The Cutoff Profile for Random Transpositions on Repeated Cards in the Full Range of Parameters
研究在重复卡片上随机交换的截断现象，通过分析总变差距离到平稳态，确定在参数全范围内的渐近形状，结合傅里叶分析和组合中心极限定理，揭示了不同情形下的高斯极限分布。
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 …
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