Analysis of top to bottom-$k$ shuffles

非对称搁架洗牌的截止现象

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。

关于随机洗牌的统计学

On the statistics of random-to-top shuffles

研究随机洗牌中固定点、下降数和翻转数的极限定理，提出新的组合分解方法，解决Pehlivan关于固定点的问题和Diaconis与Fulman关于翻转数的问题。

On card guessing game with one time riffle shuffle and complete feedback

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.

Partial mixing of semi-random transposition shuffles

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 $\leαn\log k$ …

单货架洗牌的位置矩阵

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。

随机洗牌的转置前2的极限分布

Limit profile for the transpose top-2 with random shuffle

本文利用非交换傅里叶分析的比较方法，研究交错群上由特定3-循环生成的随机洗牌游走，确定其极限分布并给出交错群图的完整谱。

Shuffles on Coxeter groups

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…

重叠循环洗牌的混合时间

Mixing Time of the Overlapping Cycles Shuffle

本文研究重叠循环洗牌的整副牌混合时间，证明其与单张牌松弛时间界仅相差 $\log(n)^3$ 因子。

星置换洗牌的击中时间混合

Hitting-time mixing for the star transposition shuffle

该研究针对对称群Sₙ上的星置换洗牌，证明其在非顶牌首次被全部选中的时刻τ渐近混合，总变分距离有上界，通过与随机置换洗牌对比的方法引入了适用于非共轭不变洗牌的技术。

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 …

不对称洗牌后的猜牌游戏

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。

块大小增长的有色顶m到随机洗牌的截止分布

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。

关于猜牌游戏：无反馈单次洗牌下的极限定律

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。

群上马尔可夫链双陪集归并的一个充要条件及其在随机置顶洗牌中的应用

A necessary and sufficient condition for double coset lumping of Markov chains on groups with an application to the random to top shuffle

本文给出群上随机游走在双陪集空间上诱导马尔可夫链的充要条件，并应用该结果分析随机置顶洗牌的一个变体，确定其不变分布、特征值及收敛速率。

关于猜牌游戏：单次洗牌（riffle shuffle）的极限定律

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。

有色top-$m$-to-random洗牌的显式截止轮廓

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。

有限群左陪集上左不变随机游走的弱归并

Weak lumping of left-invariant random walks on left cosets of finite groups

本文刻画了有限群左陪集上左不变随机游走弱归并的初始分布与权重条件，给出阿贝尔子群情形下的线性方程组判据，并应用于洗牌模型中顶牌序列的马尔可夫性验证。