Related papers: Cutoff for generalised Bernoulli-Laplace urn model…
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…
We resolve the long-standing problem of elucidating the cutoff phenomenon for a vast and important class of Markov processes, namely Markov diffusions with non-negative Bakry-\'Emery curvature. More precisely, we prove that any sequence of…
We study several exactly solvable Polya-Eggenberger urn models with a \emph{diminishing} character, namely, balls of a specified color, say $x$ are completely drawn after a finite number of draws. The main quantity of interest here is the…
We investigate a quadratic dynamical system known as nonlinear recombinations. This system models the evolution of a probability measure over the Boolean cube, converging to the stationary state obtained as the product of the initial…
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…
Low-dimensional dynamical systems are fruitful models for mixing in fluid and granular flows. We study a one-dimensional discontinuous dynamical system (termed "cutting and shuffling" of a line segment), and we present a comprehensive…
P\'olya urns are urns where at each unit of time a ball is drawn and replaced with some other balls according to its colour. We introduce a more general model: the replacement rule depends on the colour of the drawn ball and the value of…
In this article, we prove the cutoff phenomenon for a general class of the discrete-time nonlinear recombination models. This system models the evolution of a probability measure on a finite product space $S^n$ representing the state of…
The cutoff phenomenon was recently shown to systematically follow from non-negative curvature and the product condition, for all Markov diffusions. The proof crucially relied on a classical \emph{chain rule} satisfied by the carr\'e du…
In this work we introduce a new type of urn model with infinite but countable many colors indexed by an appropriate infinite set. We mainly consider the indexing set of colors to be the $d$-dimensional integer lattice and consider balanced…
An urn model of Diaconis and some generalizations are discussed. A convergence theorem is proved that implies for Diaconis' model that the empirical distribution of balls in the urn converges with probability one to the uniform…
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…
We study a P\'olya-type urn model defined as follows. Start at time 0 with a single ball of some colour. Then, at each time n>0, choose a ball from the urn uniformly at random. With probability 1/2<p<1, return the ball to the urn along with…
We analyze a Markov chain, known as the product replacement chain, on the set of generating $n$-tuples of a fixed finite group $G$. We show that as $n \rightarrow \infty$, the total-variation mixing time of the chain has a cutoff at time…
We collect, survey and develop methods of (one-dimensional) stochastic approximation in a framework that seems suitable to handle fairly broad generalizations of Polya urns. To show the applicability of the results we determine the limiting…
We consider multicolor urn models with multiple drawings. An urn model is called linear if the conditional expected value of the urn composition at time $n$ is a linear function of the composition at time $n-1$. For four different sampling…
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…
Monotonic surfaces spanning finite regions of $Z^d$ arise in many contexts, including DNA-based self-assembly, card-shuffling and lozenge tilings. One method that has been used to uniformly generate these surfaces is a Markov chain that…
Consider an urn containing balls labeled with integer values. Define a discrete-time random process by drawing two balls, one at a time and with replacement, and noting the labels. Add a new ball labeled with the sum of the two drawn…
Consider two urns, $A$ and $B$, where initially $A$ contains a large number $n$ of balls and $B$ is empty. At each step, with equal probability, either we pick a ball at random in $A$ and place it in $B$, or vice-versa (provided of course…