Related papers: Cutoff for product replacement on finite groups
Suppose X and Y are two independent irreducible Markov chains on n states. We consider the intersection time, which is the first time their trajectories intersect. We show for reversible and lazy chains that the total variation mixing time…
Let $G$ be a finite group generated by $k$ elements. The well-known product replacement algorithm provides an effective method for sampling generating sets of $G$. We study a refinement of this algorithm that is designed to output…
In the present paper, we consider a class of Markov processes on the discrete circle which has been introduced by K\"onig, O'Connell and Roch. These processes describe movements of exchangeable interacting particles and are discrete…
Consider the random Cayley graph of a finite group $G$ with respect to $k$ generators chosen uniformly at random, with $1 \ll \log k \ll \log |G|$ (ie $1 \ll k = |G|^{o(1)}$). A conjecture of Aldous and Diaconis (1985) asserts, for…
We examine the mixing time for random walks on graphs. In particular we are interested on investigating graphs with bottlenecks. Furthermore, the cutoff phenomenon is examined.
We present new concentration of measure inequalities for Markov chains, generalising results for chains that are contracting in Wasserstein distance. These are particularly suited to establishing the cut-off phenomenon for suitable chains.…
We study the mixing time of a systematic scan Markov chain for sampling from the uniform distribution on proper 7-colourings of a finite rectangular sub-grid of the infinite square lattice, the grid. A systematic scan Markov chain cycles…
A deck of $n$ cards are shuffled by repeatedly taking off the top card, flipping it with probability $1/2$, and inserting it back into the deck at a random position. This process can be considered as a Markov chain on the group $B_n$ of…
We estimate the mixing time of the a nonreversible finite Markov chain called Repeated Balls-into-Bins (RBB) process. This process is a discrete time conservative interacting particle system with parallel updates. Place initially in $L$…
This paper considers the $(n,k)$-Bernoulli--Laplace model in the case when there are two urns, the total number of red and white balls is the same, and the number of selections $k$ at each step is on the same asymptotic order as the number…
The problem of efficiently sampling from a set of(undirected) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The…
We consider Glauber dynamics for the Ising model on the complete graph on $n$ vertices, known as the Curie-Weiss model. It is well-known that the mixing-time in the high temperature regime ($\beta < 1$) has order $n\log n$, whereas the…
We obtain a tight bound of $O(L^2\log k)$ for the mixing time of the exclusion process in $\mathbf{Z}^d/L\mathbf{Z}^d$ with $k\leq{1/2}L^d$ particles. Previously the best bound, based on the log Sobolev constant determined by Yau, was not…
We address the problem of estimating the mixing time of a Markov chain from a single trajectory of observations. Unlike most previous works which employed Hilbert space methods to estimate spectral gaps, we opt for an approach based on…
Let {X_n,n\geq0} be a Markov chain on a general state space X with transition probability P and stationary probability \pi. Suppose an additive component S_n takes values in the real line R and is adjoined to the chain such that…
How many shuffles are needed to mix up a deck of cards? This question may be answered in the language of a random walk on the symmetric group, $S_{52}$. This generalises neatly to the study of random walks on finite groups, themselves a…
We analyze the mixing behavior of the biased exclusion process on a path of length $n$ as the bias $\beta_n$ tends to $0$ as $n \to \infty$. We show that the sequence of chains has a pre-cutoff, and interpolates between the unbiased…
Through a Metropolis-like algorithm with single step computational cost of order one, we build a Markov chain that relaxes to the canonical Fermi statistics for k non-interacting particles among m energy levels. Uniformly over the…
The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the…
Dealing with finite Markov chains in discrete time, the focus often lies on convergence behavior and one tries to make different copies of the chain meet as fast as possible and then stick together. There is, however, a very peculiar kind…