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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)

In a famous paper 8] Hammersley investigated the length L n of the longest increasing subsequence of a random n-permutation. Implicit in that paper is a certain one-dimensional continuous-space interacting particle process. By studying a hydrodynamical limit for Hammersley's process we show by fairly \soft" arguments that limn ?1=2 EL n = 2. This is a known… (More)

We develop a clear connection between deFinetti's theorem for exchangeable arrays (work of Aldous–Hoover–Kallenberg) and the emerging area of graph limits (work of Lovász and many coauthors). Along the way, we translate the graph theory into more classical probability .

- Dave Bayer, Persi Diaconis
- 2008

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Institute of… (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 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)

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)

- P. Diaconis
- 1996

This paper develops bounds on the rate of decay of powers of Markov kernels on finite state spaces. These are combined with eigenvalue estimates to give good bounds on the rate of convergence to stationarity for finite Markov chains whose underlying graph has moderate volume growth. Roughly, for such chains, order (diameter) 2 steps are necessary and… (More)