Related papers: Mixing Time of the Rudvalis Shuffle

We study mixing times of the one-sided $k$-transposition shuffle. We prove that this shuffle mixes relatively slowly, even for $k$ big. Using the recent "lifting eigenvectors" technique of Dieker and Saliola and applying the $\ell^2$ bound,…

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…

The random transposition shuffle on repeated cards induces a Markov chain on the quotient space of arrangements with multiplicities, and is equivalent to the many-urn mean-field Bernoulli-Laplace model introduced by Scarabotti. Writing…

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…

This paper studies Markov chains on the symmetric group $S_n$ where the transition probabilities are given by the Ewens distribution with parameter $\theta>1$. The eigenvalues are identified to be proportional to the content polynomials of…

Let $P$ be a bistochastic matrix of size $n$, and let $\Pi$ be a permutation matrix of size $n$. In this paper, we are interested in the mixing time of the Markov chain whose transition matrix is given by $Q=P\Pi$. In other words, the chain…

We prove tight bounds on the relaxation time of the so-called $L$-reversal chain, which was introduced by R. Durrett as a stochastic model for the evolution of chromosome chains. The process is described as follows. We have $n$ distinct…

Consider a permutation $\sigma\in S_n$ as a deck of cards numbered from 1 to $n$ and laid out in a row, where $\sigma_j$ denotes the number of the card that is in the $j$-th position from the left.\rm\ We define two cyclic to random…

Let 0<\alpha<1/2. We show that the mixing time of a continuous-time reversible Markov chain on a finite state space is about as large as the largest expected hitting time of a subset of stationary measure at least \alpha of the state space.…

This paper originally showed a lower bound on mixing time for a non-reversible Markov chain in terms of its largest non-trivial eigenvalue, and used this to re-derive some generalizations of results of Fan Chung. However, the paper has been…

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…

The motivation of this work is to extend the techniques of higher order random walks on simplicial complexes to analyze mixing times of Markov chains for combinatorial problems. Our main result is a sharp upper bound on the second…

We prove an upper bound on the total variation mixing time of a finite Markov chain in terms of the absolute spectral gap and the number of elements in the state space. Unlike results requiring reversibility or irreducibility, this bound is…

We develop Markov chain mixing time estimates for a class of Markov chains with restricted transitions. We assume transitions may occur along a cycle of $n$ nodes and on $n^\gamma$ additional edges, where $\gamma < 1$. We find that the…

Consider a Markov chain with finite state space and suppose you wish to change time replacing the integer step index $n$ with a random counting process $N(t)$. What happens to the mixing time of the Markov chain? We present a partial reply…

We analyze the mixing time of a popular shuffling machine known as the shelf shuffler. It is a modified version of a $2m$-handed riffle shuffle ($m=10$ in casinos) in which a deck of $n$ cards is split multinomially into $2m$ piles, the…

We present a Markov chain example where non-reversibility and an added edge jointly improve mixing time: when a random edge is added to a cycle of $n$ vertices and a Markov chain with a drift is introduced, we get mixing time of…

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…

We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees.

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…