Optimal strong stationary times for random walks on the chambers of a hyperplane arrangement

This paper studies Markov chains on the chambers of real hyperplane arrangements, a model that generalizes famous examples, such as the Tsetlin library and riffle shuffles. We discuss cutoff for the Tsetlin library for general weights, and we give an exact formula for the separation distance for the hyperplane arrangement walk. We introduce lower bounds, which allow for the first time to study cutoff for hyperplane arrangement walks under certain conditions. Using similar techniques, we also prove a uniform lower bound for the mixing time of Glauber dynamics on a monotone system.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1605.0833

A rule of thumb for riffle shuffling

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 log_2(n). We derive closed formulae and an asymptotic `rule of thumb' formula which is remarkably accurate.Comment: 27 pages, 5 table

Cutoff Phenomenon for Random Walks on Kneser Graphs

The cutoff phenomenon for an ergodic Markov chain describes a sharp transition in the convergence to its stationary distribution, over a negligible period of time, known as cutoff window. We study the cutoff phenomenon for simple random walks on Kneser graphs, which is a family of ergodic Markov chains. Given two integers $n$ and $k$, the Kneser graph $K(2n+k,n)$ is defined as the graph with vertex set being all subsets of $\{1,\ldots,2n+k\}$ of size $n$ and two vertices $A$ and $B$ being connected by an edge if $A\cap B =\emptyset$. We show that for any $k=O(n)$, the random walk on $K(2n+k,n)$ exhibits a cutoff at $\frac{1}{2}\log_{1+k/n}{(2n+k)}$ with a window of size $O(\frac{n}{k})$

Analysis of casino shelf shuffling machines

Many casinos routinely use mechanical card shuffling machines. We were asked to evaluate a new product, a shelf shuffler. This leads to new probability, new combinatorics and to some practical advice which was adopted by the manufacturer. The interplay between theory, computing, and real-world application is developed.Comment: Published in at http://dx.doi.org/10.1214/12-AAP884 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The cutoff phenomenon for finite Markov Chains

A card player may ask the following question: how many shuffles are needed to mix up a deck of cards? Mathematically, this question falls in the realm of the quantitative study of the convergence of finite Markov chains. Similar convergence rate questions for finite Markov chains are important in many fields including statistical physics, computer science, biology and more. In this dissertation, we discuss a behavior ---the cutoff phenomenon--- that is known to appear in many models. For these models, after a waiting period, the chain abruptly converges to its stationary distribution. Our aim is to develop a theory of this phenomenon and to illustrate this theory with interesting examples. We focus on the case when the convergence is measured at the $\ell^p$-distance for $1\le p\le\infty$. For $p=1$, one recovers the classical total variation distance. One of the main result of the thesis is that for families of reversible Markov chains and $1<p\le\infty$, the existence of an $\ell^p$-cutoff can be characterized using two parameters: the spectral gap and the mixing time. This fails when $p=1$. The notion of cutoff for a family of Markov chains indexed by $n$ involves a cutoff time sequence $(t_n)_1^\infty$ and window size sequence $(b_n)_1^\infty$. Ideally, when a cutoff exists, we would like to determine precisely $t_n$ and $b_n$. When $p=2$, spectral theory allows for a deeper analysis of the cutoff phenomenon producing in some cases the asymptotic behavior of the sequences $(t_n)_1^\infty$ and $(b_n)_1^\infty$. Throughout the thesis, examples are provided to illustrate the theoretical results. In particular, the last chapter is devoted to the study of the cutoff for the randomized riffle shuffle

Mixing by Cutting and Shuffling a Line Segment: The Effect of Incorporating Diffusion

Dynamical systems are commonly used to model mixing in fluid and granular flows. We consider a one-dimensional discontinuous dynamical system model (termed “cutting and shuffling” of a line segment), and we present a comprehensive computational study of finite-time mixing. The properties of the system depend on several parameters in a sensitive way, and the effect of each parameter is examined. Space-time and waterfall plots are introduced to visualize the mixing process with different mixing protocols without diffusion, showing a variety of distinct and complex behaviors in this “simple” dynamical system. To improve the mixing efficiency and avoid pathological cases, we incorporate diffusion into this model dynamical system. We show that diffusion can be quite effective at homogenizing a “mixture.” To make this effect clear, we compare cases without diffusion to those with “small” diffusivity and “large” diffusivity. Illustrative examples also show how to adapt mixing metrics from the literature, namely the number of cutting interfaces and a mixing norm, to quantify the degree of mixing in our cutting and shuffling system. To study the evolution of mixing through a large set of possible cutting and shuffling parameters, we introduce fit functions for the number of cutting interfaces and the mixing norm. These fits allow us to determined time constants of mixing for each different system considered, thereby quantifying the “speed” of mixing. Systems with various different permutations (shuffling protocols) are considered, then average properties can be computed, which hold true (on average) for all allowed permutations. The relationship between the fit parameters and the system parameters is also investigated through scatter plots in the fit parameter space. Next, universal mixing behaviors are identified by specifically introducing a critical half-mixing time, which must be found computationally. Using the latter, a rescaling of different dynamical regimes (decay curves of the mixing norm) fall onto a universal profile valid across all parameters of the cutting and shuffling dynamical system. Then, a prediction for this critical half-mixing time is made on the basis of the evolution of the number of subsegments of continuous color (unmixed subsegments). This prediction, which is called a stopping time in the finite Markov chain literature, must also be found numerically. The latter compares well with the previously computed half-mixing time, which provides an approach to determine when a system has become uniform. Finally, we examine the dependence of the half-mixing times on the characteristic P´eclet number of the system (an inverse dimensionless diffusivity), and we show that as the P´eclet number becomes large, the system transitions more sharply from an unmixed initial state to a mixed final state. This phenomenon, which is know as a “cut-off” in the finite Markov chain literature thus appears to be well substantiated by our numerical investigation of cutting and shuffling a line segment in the presence of diffusion