Related papers: Mixing time of the torus shuffle

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…

Consider a randomly shuffled deck of $2n$ cards with $n$ red cards and $n$ black cards. We study the average number of moves it takes to go from a randomly shuffled deck to a deck that alternates in color by performing the following move:…

We consider a Markov chain on invertible $n\times n$ matrices with entries in $\mathbb{Z}_2$ which moves by picking an ordered pair of distinct rows and add the first one to the other, modulo $2$. We establish a logarithmic Sobolev…

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…

A deck of $n$ cards is shuffled by repeatedly moving the top card to one of the bottom $k_n$ positions uniformly at random. We give upper and lower bounds on the total variation mixing time for this shuffle as $k_n$ ranges from a constant…

To find the number of assignments of zeros and ones satisfying a specific Knapsack Problem is $\#P$ hard, so only approximations are envisageable. A Markov chain allowing uniform sampling of all possible solutions is given by Luby, Randall…

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…

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…

We consider tilings of $\mathbb{Z}^2$ by two types of squares. We are interested in the rate of convergence to the stationarity of a natural Markov chain defined for square tilings. The rate of convergence can be represented by the mixing…

Consider an n by n array of cards shuffled in the following manner. An element x of the array is chosen uniformly at random; Then with probability 1/2 the rectangle of cards above and to the left of x is rotated 180 degrees, and with…

Consider shuffling a deck of $n$ cards, labeled $1$ through $n$, as follows: at each time step, pick one card uniformly with your right hand and another card, independently and uniformly with your left hand; then swap the cards. How long…

Establishing cutoff, an abrupt transition from "not mixed" to "well mixed", is a classical topic in the theory of mixing times for Markov chains. Interest has grown recently in determining not only the existence of cutoff and the order of…

Mixing of finite time-homogeneous Markov chains is well understood nowadays, with a rich set of techniques to estimate their mixing time. In this paper, we study the mixing time of random walks in dynamic random environments. To that end,…

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…

In this paper, we study the biased random transposition shuffle, a natural generalization of the classical random transposition shuffle studied by Diaconis and Shahshahani. We diagonalize the transition matrix of the shuffle and use these…

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…

The switch chain is a well-known Markov chain for sampling directed graphs with a given degree sequence. While not ergodic in general, we show that it is ergodic for regular degree sequences. We then prove that the switch chain is rapidly…

We consider a family of card shuffles of $n$ cards in which the allowed moves involve transpositions corresponding to the Jucys--Murphy elements of the symmetric group $\{S_m\}_{m \leq n}$. We determine the eigenvalues of the corresponding…

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…

A Gilbert-Shannon-Reeds (GSR) shuffle is performed on a deck of $N$ cards by cutting the top $n\sim Bin(N,1/2)$ cards and interleaving the two resulting piles uniformly at random. The celebrated "Seven shuffles suffice" theorem of…