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MIXING

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by Dana Randall
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BibTeX

@MISC{Randall_mixing,
    author = {Dana Randall},
    title = {MIXING},
    year = {}
}

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Abstract

In this tutorial, we introduce the notion of a Markov chain and explore how it can be used to sample from a large set of configurations. Our primary focus is determining how quickly a Markov chain “mixes,” or converges to its stationary distribution, as this is the key factor in the running time. We provide an overview of several techniques used to establish good bounds on the mixing time. The applications are mostly chosen from statistical physics, although the methods are much more general.

Citations

1648 Equation of state calculations by fast computing machines - Metropolis, Rosenbluth, et al. - 1953
409 The complexity of computing the permanent - Valiant - 1979
317 Reversible Markov chains and random walks on graphs,” unpublished manuscript - Aldous, Fill
257 Approximating the Permanent - Jerrum, Sinclair - 1989
253 A PolynomialTime Approximation Algorithm for the Permanent of a Matrix with Nonnegative Entries - Jerrum, Sinclair, et al.
250 Random Generation of Combinatorial Structures from a Uniform Distributions, Theoret. Computer Sci - Jerrum, Valiant, et al. - 1986
216 Approximate counting, uniform generation and rapidly mixing Markov chains - Sinclair, Jerrum - 1989
203 The Markov chain Monte Carlo method: An approach to approximate counting and integration - Jerrum, Sinclair - 1996
200 Random walk on finite groups and rapidly mixing Markov chains - Aldous - 1983
166 Algorithms for Random Generation and Counting: A Markov Chain Approach, Birkhäuser - Sinclair - 1993
158 Improved bounds for mixing rates of Markov chains and multicommodity flow - Sinclair - 1992
136 Comparison theorems for reversible Markov chains - Diaconis, Saloff-Coste - 1993
130 A random polynomial time algorithm for approximating the volume of a convex body - Dyer, Frieze, et al. - 1991
58 Analyzing Glauber dynamics by comparison of Markov chains - Randall, Tetali
55 Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem - Salas, Sokal - 1997
53 Faster random generation of linear extensions - Bubley, Dyer - 1998
48 Markov chains and polynomial time algorithms - Kannan - 1994
42 Random walks on truncated cubes and sampling 0-1 knapsack problem - Morris, Sinclair - 1999
36 A more rapidly mixing Markov chain for graph colorings - Dyer, Greenhill - 1998
36 Fast convergence of the Glauber dynamics for sampling independent sets, Random Struct. Algorithms 15(3-4 - Luby, Vigoda - 1999
34 Markov chain decomposition for convergence rate analysis - Madras, Randall
30 A non-Markovian coupling for randomly sampling colorings, PDF file available - Hayes, Vigoda - 2003
30 Mixing times, Microsurveys in discrete probability - Lovász, Winkler - 1997
29 The Metropolis algorithm - Beichl, Sullivan
29 The Glauber dynamics on colorings of a graph with high girth and maximum degree - MOLLOY
27 An introduction to the Ising model - Cipra - 1987
27 Sampling adsorbing staircase walks using a new Markov chain decomposition method - Martin, Randall - 2000
20 An extension of path coupling and its application to the Glauber dynamics for graph colourings - Dyer, Goldberg, et al.
9 Markov chains for planar lattice structures - Luby, Randall, et al.
9 Sampling spin configurations of an ising system - Randall, Wilson - 1999
7 Disjoint decomposition with applications to sampling circuits in some Cayley graphs - Martin, Randall - 2003
6 Convergence rates for Monte Carlo experiments. Numerical Methods for Polymeric Systems - Sinclair - 1997
4 A very simple algorithm for estimating the number of k -colorings of a low-degree graph. Random Structures and Algorithms - Jerrum - 1995
3 Generating random spanning trees. em - Broder - 1989
2 Decomposition Methods and Sampling Circuits in the Cartesian Lattice - Randall
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