Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics

Borgs, Christian; Chayes, Jennifer; Frieze, Alan; Kim, Jeong Han; Tetali, Prasad; Vigoda, Eric; Vu, Van

doi:10.1109/sffcs.1999.814594

Cited by 85 publications

(158 citation statements)

References 25 publications

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“…al. [7] studied the Swendsen-Wang and the Potts model near their phase transitions, and by constructing a set A where the conductance Φ(A) is exponentially small they were able to establish that both Markov chains 298…”

Section: A Geometric Lower Boundmentioning

confidence: 99%

Mathematical Aspects of Mixing Times in Markov Chains

Montenegro

Tetali

2006

FNT in Theoretical Computer Science

209

222

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In the past few years we have seen a surge in the theory of finite Markov chains, by way of new techniques to bounding the convergence to stationarity. This includes functional techniques such as logarithmic Sobolev and Nash inequalities, refined spectral and entropy techniques, and isoperimetric techniques such as the average and blocking conductance and the evolving set methodology. We attempt to give a more or less self-contained treatment of some of these modern techniques, after reviewing several preliminaries. We also review classical and modern lower bounds on mixing times. There have been other important contributions to this theory such as variants on coupling techniques and decomposition methods, which are not included here; our choice was to keep the analytical methods as the theme of this presentation. We illustrate the strength of the main techniques by way of simple examples, a recent result on the Pollard Rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.

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Section: A Geometric Lower Boundmentioning

confidence: 99%

Mathematical Aspects of Mixing Times in Markov Chains

Montenegro

Tetali

2006

FNT in Theoretical Computer Science

209

222

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“…As a result, in order to support an arrival rate of λ = 0.5 − , > 0, the upper-bound on the delay can grow as fast as (1/ ) L . Furthermore, for the case where is sufficiently small, a lower-bound on the delay has been derived that grows exponentially with exponent √ L/(log L) 2 [10]. These bounds imply that the delay performance of CSMA becomes very poor for large networks, i.e.…”

Section: Highlights Of Technical Resultsmentioning

confidence: 98%

Throughput-delay trade-off of CSMA policies in wireless networks

Lotfinezhad

Marbach

2010

IEEE Information Theory Workshop 2010 (ITW 2010)

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Abstract-We consider CSMA policies for multihop wireless networks. CSMA policies are simple policies that can be easily implemented in a distributed manner. However, the delay performance of CSMA policies can be very poor as the delay can grow exponentially in the network size. As a result, CSMA policies are not practical for delay-sensitive traffic even for midsized networks. In this paper, we consider a slight variant of the classical CSMA policies and show that it leads to a much improved delay performance. In particular, we show that the delay does not depend on the networks size. Using this result, we also characterize the delay-throughput trade-off of the proposed CSMA policy. At the heart of our analysis is a result that shows that CSMA policies quickly converge to a maximum schedule, i.e., converge to a maximum schedule at a rate that does not depend on the network size. Using this insight, we consider a CSMA policy that periodically "unlocks" the transmission pattern of a CSMA policy, and show that this unlocking mechanism can be used to obtain a much improved delay performance without significantly reducing the throughput. While our analysis has been carried out for the special case of an interference graph with a grid (lattice) topology, we provide numerical case studies for general network topologies and show that the intuition obtained from the analysis carries over to these general cases. We also illustrate the performance of the proposed CSMA policy when combined with a flow control mechanism.

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“…The key tools in our proofs are careful Peierls arguments, used in statistical physics to study uniqueness of the Gibbs state and phase transitions (see, e.g., [4], [6]), and in computer science to study slow mixing of Markov chains (see, e.g., [2], [10], [15]). Peierls arguments allow you to add and remove contours by complementing the interiors of those contours.…”

Section: Introductionmentioning

confidence: 99%

Clustering in Interfering Binary Mixtures

Miracle

Randall

Streib

2011

Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques

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Abstract. Colloids are binary mixtures of molecules with one type of molecule suspended in another. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters. We characterize the high and low density phases for a general family of discrete interfering binary mixtures by showing that they exhibit a "clustering property" at high density and not at low density. The clustering property states that there will be a region that has very high area to perimeter ratio and very high density of one type of molecule. A special case is mixtures of squares and diamonds on Z 2 which corresond to the Ising model at fixed magnetization.

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Torpid mixing of some Monte Carlo Markov chain algorithms in statistical physics

Cited by 85 publications

References 25 publications

Mathematical Aspects of Mixing Times in Markov Chains

Mathematical Aspects of Mixing Times in Markov Chains

Throughput-delay trade-off of CSMA policies in wireless networks

Clustering in Interfering Binary Mixtures

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