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A bstract. Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a(More)
We describe a simple one-person card game, patience sorting. Its analysis leads to a broad circle of ideas linking Young tableaux with the longest increasing subsequence of a random permutation via the Schensted correspondence. A recent highlight of this area is the work of Baik-Deift-Johansson which yields limiting probability laws via hard analysis of(More)
is an expository paper on the use of logarithmic Sobolev inequalities for bounding rates of convergence of Markov chains on finite state spaces to their stationary distributions. Logarithmic Sobolev inequalities complement eigenvalue techniques and work for nonreversible chains in continuous time. Some aspects of the theory simplify considerably with finite(More)
We analyze the most commonly used method for shuffling cards. The main result is a simple expression for the chance of any arrangement after any number of shuffles. This is used to give sharp bounds on the approach to randomness: $ log, n + 0 shuffles are necessary and sufficient to mix up n cards. Key ingredients are the analysis of a card trick and the(More)
Author names in alphabetical order. Abstract We consider a symmetric random walk on a connected graph, where each edge is labeled with the probability of transition between the two adjacent vertices. The associated Markov chain has a uniform equilibrium distribution; the rate of convergence to this distribution, i.e., the mixing rate of the Markov chain, is(More)
We consider a Markov process on a connected graph, with edges labeled with transition rates between the adjacent vertices. The distribution of the Markov process converges to the uniform distribution at a rate determined by the second smallest eigenvalue λ 2 of the Laplacian of the weighted graph. In this paper we consider the problem of assigning(More)
We give a new proof of the strong Szegö limit theorem estimating the determinants of Toeplitz matrices using symmetric function theory. We also obtain asymptotics for Toeplitz minors. If f(t)=;. −. d n t n is a function on the unit circle T in C then D n − 1 (f) will denote the Toeplitz determinant det T n − 1 (f), where T n − 1 (f) is the n × n Toeplitz(More)
Random graphs with a given degree sequence are a useful model capturing several features absent in the classical Erd˝ os-Rényi model, such as dependent edges and non-binomial degrees. In this paper, we use a characterization due to Erd˝ os and Gallai to develop a sequential algorithm for generating a random labeled graph with a given degree sequence. The(More)