Guan-Yu Chen, Laurent Saloff-Coste
We study the cutoff phenomenon for generalized riffle shuffles where, at each step, the deck of cards is cut into a random number of packs of multinomial sizes which are then riffled together.
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The well-known Gilbert-Shannon-Reeds model for riffle shuffles assumes that the cards are initially cut 'about in half' and then riffled together. We analyze a natural variant where the initial cut is biased. Extending results of Fulman (1998), we show a sharp cutoff in separation and L-infinity distances. This analysis is possible due to the close connection between shuffling and quasisymmetric functions along with some complex analysis of a generating function.
In the Gilbert-Shannon-Reeds shuffle, a deck of $N$ cards is cut into two approximately equal parts which are then riffled uniformly at random. Bayer and Diaconis famously showed that this Markov chain undergoes cutoff in total variation after $\frac{3\log(N)}{2 \log(2)}$ shuffles. We establish cutoff for the more general asymmetric riffle shuffles in which one cuts the deck into differently sized parts before riffling. The value of the cutoff point confirms a conjecture of Lalley from 2000. Some appealing consequences are that asymmetry always slows mixing and that total variation mixing is s…
This paper studies the game of guessing riffle-shuffled cards with complete feedback. A deck of $n$ cards labelled 1 to $n$ is riffle-shuffled once and placed on a table. A player tries to guess the cards from top and is given complete feedback after each guess. The goal is to find the guessing strategy with maximum reward (expected number of correct guesses). We give the optimal strategy for this game and prove that the maximum expected reward is $n/2+\sqrt{2/π}\cdot\sqrt{n}+O(1)$, partially solving an open problem of Bayer and Diaconis.
We consider a generalized riffle shuffle on the colored permutation group $G_{p, n}$ and derive a determinantal formula for the probability of finding descents at given positions, proof of which is based on the bijection between the set of shuffles in question and that of non-intersecting lattice paths.
This paper concerns a probability distribution on the symmetric group generalizing the riffle shuffle of Bayer, Diaconis, and others. There are close connections with the theory of quasisymmetric and symmetric functions.
As a continuation to our previous work, we consider a generalization of carries process. Our results are : (i) right eigenvectors of the transition probability matrix, (ii) correlation of carries between different steps, and (iii) generalized riffle shuffle having the same distribution as that of the generalized carries process.
Card guessing after an asymmetric riffle shuffle
本文研究了在不对称洗牌后如何通过最优策略最大化正确猜牌数，并分析了猜牌数的分布特性。
On Card guessing games: limit law for no feedback one-time riffle shuffle
研究无反馈猜牌游戏中单次riffle洗牌后正确猜测次数的极限行为，给出了极限定律并证明了整数矩收敛。
We study the Gilbert-Shannon-Reeds model for riffle shuffles and ask 'How many times must a deck of cards be shuffled for the deck to be in close to random order?'. In 1992, Bayer and Diaconis gave a solution which gives exact and asymptotic results for all decks of practical interest, e.g. a deck of 52 cards. But what if one only cares about the colors of the cards or disregards the suits focusing solely on the ranks? More generally, how does the rate of convergence of a Markov chain change if we are interested in only certain features? Our exploration of this problem takes us through random …
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.
By a well-known result of Bayer and Diaconis, the maximum entropy model of the common riffle shuffle implies that the number of riffle shuffles necessary to mix a standard deck of 52 cards is either 7 or 11--with the former number applying when the metric used to define mixing is the total variation distance and the latter when it is the separation distance. This and other related results assume all 52 cards in the deck to be distinct and require all $52!$ permutations of the deck to be almost equally likely for the deck to be considered well mixed. In many instances, not all cards in the deck…
On Card guessing games: limit law for one-time riffle shuffle
研究一次riffle洗牌后的猜牌游戏，给出正确猜中次数的极限定律，并建立与双色猜牌游戏的关联，进而得到纯运气猜测首次出现的极限分布。
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。
Type A affine shuffles are compared with riffle shuffles followed by a cut. Although these probability measures on the symmetric group S_n are different, they both satisfy a convolution property. Strong evidence is given that when the underlying parameter $q$ satisfies $gcd(n,q-1)=1$, the induced measures on conjugacy classes of the symmetric group coincide. This gives rise to interesting combinatorics concerning the modular equidistribution by major index of permutations in a given conjugacy class and with a given number of cyclic descents. It is proved that the use of cuts does not speed up …
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.
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。
We study the cyclic adjacent transposition (CAT) shuffle of $n$ cards, which is a systematic scan version of the random adjacent transposition (AT) card shuffle. In this paper, we prove that the CAT shuffle exhibits cutoff at $\frac{n^3}{2 π^2} \log n$, which concludes that it is twice as fast as the AT shuffle.
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…
In this thesis we introduce a new type of card shuffle called the one-sided transposition shuffle. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff…
Cutoff for the Adjacent Transposition Shuffle on a Cycle
研究环上相邻转置洗牌的混合时间，证明其在 $n^2\log n/(8\pi^2)$ 时刻出现全变差截止，窗口为 $O(n^2\log\log n)$。
We consider a random walk on the hyperoctahedral group $B_n$ generated by the signed permutations of the forms $(i,n)$ and $(-i,n)$ for $1\leq i\leq n$. We call this the flip-transpose top with random shuffle on $B_n$. We find the spectrum of the transition probability matrix for this shuffle. We prove that the mixing time for this shuffle is of order $n\log n$. We also show that this shuffle exhibits the cutoff phenomenon. In the appendix, we show that a similar random walk on the demihyperoctahedral group $D_n$ also has a cutoff at $\left(n-\frac{1}{2}\right)\log n$.
In this paper, we investigate the properties of a random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(i,n-1,n)$ and $(i,n,n-1)$. We call this the transpose top-$2$ with random shuffle. We find the spectrum of the transition matrix of this shuffle. We show that the mixing time is of order $\left(n-\frac{3}{2}\right)\log n$ and prove that there is a total variation cutoff for this shuffle.
We show that for any semi-random transposition shuffle on $n$ cards, the mixing time of any given $k$ cards is at most $n\log k$, provided $k=o((n/\log n)^{1/2})$. In the case of the top-to-random transposition shuffle we show that there is cutoff at this time with a window of size O(n), provided further that $k\to\infty$ as $n\to\infty$ (and no cutoff otherwise). For the random-to-random transposition shuffle we show cutoff at time $(1/2)n\log k$ for the same conditions on $k$. Finally, we analyse the cyclic-to-random transposition shuffle and show partial mixing occurs at time $\leαn\log k$ …
Cutoff for the Biased Random Transposition Shuffle
本文研究有偏随机对换洗牌，通过对转移矩阵对角化，证明其在特定时间呈现全变差截止现象，且截止时间附近固定牌数量的极限分布为 Poisson 分布。
Cutoff phenomenon for the warp-transpose top with random shuffle
本文研究完全单项群$G_n\wr S_n$上带随机洗牌的warp-transpose陀螺随机游走，求出其转移矩阵谱，证明混合时间为$O(n\log n+\frac{1}{2}n\log (|G_n|-1))$，并揭示其$\ell^2$截止与全变差截止现象。
In this paper, we use the eigenvalues of the random to random card shuffle to prove a sharp upper bound for the total variation mixing time. Combined with the lower bound due to Subag, we prove that this walk exhibits cutoff at $\frac{3}{4} n \log n - \frac{1}{4}n\log\log{n}$ with window of order $n$, answering a conjecture of Diaconis.
In this paper we study the mixing time of a biased transpositions shuffle on a set of $N$ cards with $N/2$ cards of two types. For a parameter $0<a \le 1$, one type of card is chosen to transpose with a bias of $\frac{a}{N}$ and the other type is chosen with probability $\frac{2-a}{N}$. We show that there is cutoff for the mixing time of the chain at time $\frac{1}{2a} N \log N$. Our proof uses a modified marking scheme motivated by Matthews' proof of a strong uniform time for the unbiased shuffle.
We introduce a new type of card shuffle called one-sided transpositions. At each step a card is chosen uniformly from the pack and then transposed with another card chosen uniformly from below it. This defines a random walk on the symmetric group generated by a distribution which is non-constant on the conjugacy class of transpositions. Nevertheless, we provide an explicit formula for all eigenvalues of the shuffle by demonstrating a useful correspondence between eigenvalues and standard Young tableaux. This allows us to prove the existence of a total-variation cutoff for the one-sided transpo…
Random Walks on the Generalized Symmetric Group: Cutoff for the One-sided Transposition Shuffle
本文基于 Matheau-Raven 的技术证明了广义对称群上无偏单侧换位洗牌在时间 $n \log(n)$ 的全变差距离和分离距离截止，并证明单模分支规则，为一般有偏情形的猜想奠定基础。
Cutoff for q-deformed classical card shuffles in the type A Iwahori--Hecke algebra
本文对角化q-变形k-星换位洗牌，并利用三种q-变形洗牌的谱证明它们在固定q>1时具有全变差和l^2截止，且改进了短系统扫描的上下界。
Cutoff for asymmetric shelf shuffle
研究非对称搁架洗牌（每张牌以概率p放在顶部或以1-p放在底部）产生的排列分布，证明下降数和谷数构成充分统计量，并给出总变差距离、分离距离和相对熵的截止现象及显式轮廓。
In this paper, we investigate the mixing time of the adjacent transposition shuffle for a deck of $N$ cards. We prove that around time $N^2\log N/(2π^2)$, the total variation distance to equilibrium of the deck distribution drops abruptly from $1$ to $0$, and that the separation distance has a similar behavior but with a transition occurring at time $(N^2\log N)/π^2$. This solves a conjecture formulated by David Wilson. We present also similar results for the exclusion process on a segment of length $N$ with $k$ particles.
Cutoff for the Overhand Shuffle
本文研究固定切割概率下翻手洗牌的全变差截止时间，证明其位于 $p^2/(2(1-p)\pi^2)n^2\log n$，并给出相对熵截止的窗口估计。
We present an overview of the representation theoretic techniques used to study the mixing times of random walks on finite groups. We focus on the card shuffle studied by Diaconis and Shahshahani in the 1980s and a recent improvement on their result by Teyssier
Cutoff profiles for colored top-m-to-random shuffles with growing block size
研究块大小m_n随n增长时，C_p≀S_n上的p-色顶-m-到-随机洗牌的截止现象，通过泊松极限和嵌套集约化得到全变差、分离和似然比分布。
Explicit cutoff profiles for colored top-$m$-to-random shuffles
研究花环积$C_p\wr S_n$上的$p$色top-$m$-to-random洗牌，利用Nakano-Sadahiro-Sakurai基元素$B_m$得到精确的嵌套集占用混合，将似然比简化为单一统计量$L_p$，从而导出分离度、$L^\infty(U)$、全变差、$L^q(U)$、$\chi^2$和相对熵的精确公式，并给出截止轮廓。
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