## Analyzing Glauber Dynamics by Comparison of Markov Chains (1999)

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Venue: | Journal of Mathematical Physics |

Citations: | 59 - 15 self |

### BibTeX

@ARTICLE{Randall99analyzingglauber,

author = {Dana Randall and Prasad Tetali},

title = {Analyzing Glauber Dynamics by Comparison of Markov Chains},

journal = {Journal of Mathematical Physics},

year = {1999},

volume = {41},

pages = {1598--1615}

}

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### Abstract

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of c...

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1476 |
Exactly Solved Models in Statistical Mechanics
- Baxter
- 1982
(Show Context)
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- Jerrum, Sinclair
- 1993
(Show Context)
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297 |
Approximating the permanent
- Jerrum, Sinclair
- 1989
(Show Context)
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281 | Random generation of combinatorial structures from a uniform distribution - Jerrum, Valiant, et al. - 1986 |

273 | Geometric bounds on eigenvalues of markov chains. Annals of applied probability
- Diaconis, Stroock
- 1991
(Show Context)
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257 |
Approximate counting, uniform generation and rapidly mixing Markov chains
- Jerrum, Sinclair
- 1989
(Show Context)
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224 |
Random walks on finite groups and rapidly mixing Markov chains. Seminar on probability
- Aldous
- 1983
(Show Context)
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180 |
Algorithms for Random Generation and Counting: A Markov Chain Approach
- Sinclair
- 1993
(Show Context)
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155 | Comparison theorems for reversible Markov chains
- Diaconis, Saloff-Coste
- 1993
(Show Context)
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- Bubley, Dyer
- 1997
(Show Context)
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123 |
Conway’s tiling groups
- Thurston
- 1990
(Show Context)
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113 | Logarithmic Sobolev Inequalities for Finite Markov Chains
- Diaconis, Saloff-Coste
- 1996
(Show Context)
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111 |
Generating a random permutation with random transpositions. Probability Theory and Related Fields
- Diaconis, Shahshahani
- 1981
(Show Context)
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109 |
A very simple algorithm for estimating the number of k-colourings of a low-degree graph
- Jerrum
- 1995
(Show Context)
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102 | Lectures on Glauber dynamics for discrete spin models. Lectures on probability and statistics (Saint-Flour
- Martinelli
- 1997
(Show Context)
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- Luby, Randall, et al.
- 1995
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Citation Context ... second significant advance is the coupling technique refined by Aldous [1], which has led to efficient approximation algorithms for colorings [13, 38], independent sets [24], and planar dimer models =-=[23]-=-, as well as problems of a more combinatorial nature (e.g., [3, 18, 26]). Further refinement, via path coupling, due to Bubley and Dyer [3] has made the coupling technique by far the most successful a... |

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- Diaconis, Saloff-Coste
- 1993
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- Kannan
- 1994
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- 1999
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- Jerrum
- 1998
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- 1997
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Percolation and the hard-core lattice gas model
- Berg, Steif
- 1994
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- McShine, Tetali
- 1999
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Possible rates of convergence in finite range, attractive spin systems
- Holley
- 1985
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- Welsh
- 1997
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- Berg, Brouwer
- 2000
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- Kenyon, Randall, et al.
- 1996
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- Molloy, Reed, et al.
- 1997
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- Madras, Randall
- 1996
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