Related papers: Card shuffling and diophantine approximation

When we want to simulate the realization of a symmetric simple random walk on $\mathbb Z^d$, we use $(2d)$-side fair dice to decide to which neighbor it jumps at each step if $d\geq 2$ or we simply use a fair coin when $d=1$. Assume that…

Let $G$ be a graph on $n$ vertices, with complement $\overline{G}$. The spectral gap of the transition probability matrix of a random walk on $G$ is used to estimate how fast the random walk becomes stationary. We prove that the larger…

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…

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}…

Juggling patterns can be described by a sequence of cards which keep track of the relative order of the balls at each step. This interpretation has many algebraic and combinatorial properties, with connections to Stirling numbers, Dyck…

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…

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…

Two notions of riffle shuffling on finite Coxeter groups are given: one using Solomon's descent algebra and another using random walk on chambers of hyperplane arrangements. These coincide for types $A$,$B$,$C$, $H_3$, and rank two groups.…

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…

We consider a natural local dynamic on the set of all rooted planar maps with $n$ edges that is in some sense analogous to "edge flip" Markov chains, which have been considered before on a variety of combinatorial structures (triangulations…

The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these…

The starting point of this work is an equality between two quantities $A$ and $B$ found in the literature, which involve the {\em doubling-modulo-an-odd-integer} map, i.e., $x\in {\mathbb N} \mapsto 2x \bmod{(2n+1)}$ for some positive…

We study the topdrop map, a mapping on permutations in $S_n$ related to card shuffling. We show this map is bijective and study its orbit structure. We introduce the notion of the topdrop-necklace as a way of classifying the orbits of the…

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…

A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$-card deck $D = \{1,...,n\}$. The permutation $\sigma$ is simply the sequence of cards in the order they are removed. This permutation…

We show that the spectral gap of a random walk on the domain of normal attraction of an $\alpha$-stable law is of order $\mathcal O(n^{\alpha})$ when restricted to boxes of size $n$. The proof is based on a comparison principle that may be…

We study a family of maps from $S_n \to S_n$ we call fixed point homing shuffles. These maps generalize a few known problems such as Conway's Topswops, and a card shuffling process studied by Gweneth McKinley. We show that the iterates of…

Consider a uniformly random deck consisting of cards labelled by numbers from $1$ through $n$, possibly with repeats. A guesser guesses the top card, after which it is revealed and removed and the game continues. What is the expected number…

Untangling is a process in which some vertices of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph C_n while keeping at…

We study an example of a {\em hit-and-run} random walk on the symmetric group $\mathbf S_n$. Our starting point is the well understood {\em top-to-random} shuffle. In the hit-and-run version, at each {\em single step}, after picking the…