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Sampling algorithms for Discrete Markov Random Fields and Related Graphical Models

JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 2021年第536期116卷 2065-2086页

作者： Izenman, Alan Julian Temple Univ Dept Stat Sci 1810 Liacouras Walk Philadelphia PA 19122 USA Temple Univ Data Sci Inst 1810 Liacouras Walk Philadelphia PA 19122 USA

Discrete Markov random fields are undirected graphical models in which the nodes of a graph are discrete random variables with values usually represented by colors. Typically, graphs are taken to be square lattices, although more general graphs are also of interest. Such discrete MRFs have been studied in many disciplines. We describe the two most popular types of discrete MRFs, namely the two-state Ising model and the q-state Potts model, and variations such as the cellular automaton, the cellular Potts model, and the random cluster model, the latter of which is a continuous generalization of both the Ising and Potts models. Research interest is usually focused on providing algorithms for simulating from these models because the partition function is so computationally intractable that statistical inference for the parameters of the appropriate probability distribution becomes very complicated. Substantial improvements to the Metropolis algorithm have appeared in the form of cluster algorithms, such as the Swendsen-Wang and wolff algorithms. We study the simulation processes of these algorithms, which update the color of a cluster of nodes at each iteration.

关键词： Cellular Potts model Ising model Metropolis algorithm Potts model Swendsen– Wang algorithm wolff algorithm

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RARITY OF EXTREMAL EDGES IN RANDOM SURFACES AND OTHER THEORETICAL APPLICATIONS OF CLUSTER algorithmS

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ANNALS OF APPLIED PROBABILITY 2020年第5期30卷 2439-2464页

作者： Cohen-Alloro, Omri Peled, Ron Tel Aviv Univ Sch Math Sci Tel Aviv Israel

Motivated by questions on the delocalization of random surfaces, we prove that random surfaces satisfying a Lipschitz constraint rarely develop extremal gradients. Previous proofs of this fact relied on reflection positivity and were thus limited to random surfaces defined on highly symmetric graphs, whereas our argument applies to general graphs. Our proof makes use of a cluster algorithm and reflection transformation for random surfaces of the type introduced by Swendsen-Wang, wolff and Evertz et al. We discuss the general framework for such cluster algorithms, reviewing several particular cases with emphasis on their use in obtaining theoretical results. Two additional applications are presented: A reflection principle for random surfaces and a proof that pair correlations in the spin O (n) model have monotone densities, strengthening Griffiths' first inequality for such correlations.

关键词： Random surfaces extremal gradients cluster algorithms Swendsen-Wang algorithm wolff algorithm

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Quantum Criticality in the Transverse Field Random Ising Model

Quantum Criticality in the Transverse Field Random Ising Mod...

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作者： Kristoffer Aronsen KTH Royal Institute of Technology

学位级别：硕士

In this thesis the critical behaviour at the quantum phase transition of a transverse field random Ising chain is studied by mapping it to corresponding two-dimensional classical model. Monte Carlo simulations using the wolff algorithm are performed in order to sample the system configurations. The critical point of the transition is located by measuring the Binder cumulant. An analysis of finite size scaling corrections is performed and it is demonstrated that a higher order correction to scaling is visible. The dynamic scaling of the system is studied, and it is shown that by taking proper account of scaling corrections the analytic prediction of activated dynamic scaling is observed. Average and typical correlations are measured for the system at the critical point and the scaling behaviour obtained is compared to analytical results. A discrepancy between the numerical and analytical value for the critical exponent of the average correlation is found. The scaling behaviour of the typical correlation is found to align well with the analytic value.

关键词： Transverse field random Ising chain Monte Carlo simulation wolff algorithm critical exponents finite size scaling quantum phase transition

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Cluster Monte Carlo and numerical mean field analysis for the water liquid-liquid phase transition

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COMPUTER PHYSICS COMMUNICATIONS 2009年第4期180卷 497-502页

作者： Mazza, Marco G. Stokely, Kevin Strekalova, Elena G. Stanley, H. Eugene Franzese, Giancarlo Boston Univ Ctr Polymer Studies Boston MA 02215 USA Boston Univ Dept Phys Boston MA 02215 USA Univ Barcelona Dept Fis Fonamental E-08028 Barcelona Spain

Using wolffs cluster Monte Carlo simulations and numerical minimization within a mean field approach, we study the low temperature phase diagram of water, adopting a cell model that reproduces the known properties of water in its fluid phases. Both methods allow us to study the thermodynamic behavior of water at temperatures, where other numerical approaches - both Monte Carlo and molecular dynamics - are seriously hampered by the large increase of the correlation times. The cluster algorithm also allows us to emphasize that the liquid-liquid phase transition corresponds to the percolation transition of tetrahedrally ordered water molecules. Published by Elsevier B.V.

关键词： wolff algorithm Liquid-liquid phase transitions

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Improving Random Number Generators in the Monte Carlo simulations via twisting and combining

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COMPUTER PHYSICS COMMUNICATIONS 2008年第6期178卷 401-408页

作者： Deng, Lih-Yuan Guo, Rui Lin, Dennis K. J. Bai, Fengshan Tsinghua Univ Dept Math Sci Beijing 100084 Peoples R China Univ Memphis Dept Math Sci Memphis TN 38152 USA Penn State Univ Dept Supply Chain & Informat Syst University Pk PA 16802 USA

Problems for various random number generators accompanying the wolff algorithm [U. wolff, Phys. Rev. Lett. 62 (1989) 361;U. wolff, Phys. Lett. B 228 (1989) 379] are discussed, including the hidden errors first reported in [A.M. Ferrenberg, D.P. Landau, Y.J. Wong, Phys. Rev. Lett. 69 (1992) 3382]. A general (though simple) method of twisting and combining for improving the performance of these generators is proposed. Some recent generators motivated by such a twisting and combining method with extremely long period are discussed. The proposed method provides a novel and simple way to improve RNGs in its performance. (C) 2007 Elsevier B.V. All rights reserved.

关键词： DX generator hidden errors LCG LFG MRG RANLUX SWC twisting and combining (TAC) wolff algorithm

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Using duration models to reduce fragmentation in audio segmentation

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MACHINE LEARNING 2006年第2-3期65卷 485-515页

作者： Abdallah, Samer Sandler, Mark Rhodes, Christophe Casey, Michael Univ London QueenMary London E1 4NS England Univ London Goldsmiths Coll London SE14 6NW England

We investigate explicit segment duration models in addressing the problem of fragmentation in musical audio segmentation. The resulting probabilistic models are optimised using Markov Chain Monte Carlo methods;in particular, we introduce a modification to wolff's algorithm to make it applicable to a segment classification model with an arbitrary duration prior. We apply this to a collection of pop songs, and show experimentally that the generated segmentations suffer much less from fragmentation than those produced by segmentation algorithms based on clustering, and are closer to an expert listener's annotations, as evaluated by two different performance measures.

关键词： segmentation duration prior MCMC Gibbs sampling wolff algorithm

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Anisotropic scaling and generalized conformal invariance at Lifshitz points

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COMPUTER PHYSICS COMMUNICATIONS 2002年第1-2期147卷 419-422页

作者： Henkel, M Pleimling, M Univ Nancy 1 Phys Mat Lab CNRS UMR 7556 F-54506 Vandoeuvre Les Nancy France Univ Erlangen Nurnberg Inst Theoret Phys 1 D-91058 Erlangen Germany

A new variant of the wolff cluster algorithm is proposed for simulating systems with competing interactions. This method is used in a high-precision study of the Lifshitz point of the 3D ANNNI model. At the Lifshitz point, several critical exponents are found and the anisotropic scaling of the correlators is verified. The functional form of the two-point correlators is shown to be consistent with the predictions of generalized conformal invariance. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： conformal invariance Lifshitz point ANNNI model correlation function wolff algorithm

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Anisotropic scaling and generalized conformal invariance at Lifshitz points

Anisotropic scaling and generalized conformal invariance at ...

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Europhysics Conference on Computational Physics (CCP 2001)

作者： Henkel, M Pleimling, M Univ Nancy 1 Phys Mat Lab CNRS UMR 7556 F-54506 Vandoeuvre Les Nancy France Univ Erlangen Nurnberg Inst Theoret Phys 1 D-91058 Erlangen Germany

关键词： conformal invariance Lifshitz point ANNNI model correlation function wolff algorithm

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Cluster Monte Carlo simulations of the nematic-isotropic transition

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Physical Review E 2001年第6期63卷 062702-062702页

作者： N. V. Priezjev Robert A. Pelcovits Department of Physics Brown University Providence Rhode Island 02912

We report the results of simulations of the three-dimensional Lebwohl-Lasher model of the nematic-isotropic transition using a single cluster Monte Carlo algorithm. The algorithm, first introduced by Kunz and Zumbach to study two-dimensional nematics, is a modification of the wolff algorithm for spin systems, and greatly reduces critical slowing down. We calculate the free energy in the neighborhood of the transition for systems up to linear size 70. We find a double well structure with a barrier that grows with increasing system size. We thus obtain an upper estimate of the value of the transition temperature in the thermodynamic limit.

关键词： .critical slowing system size thermodynamic limit LL model Lebwohl-Lasher single cluster wolff algorithm Kunz and Zumbach NI transition

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Critical dynamics behavior of the wolff algorithm in the site-bond-correlated Ising model

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INTERNATIONAL JOURNAL OF MODERN PHYSICS C 1998年第6期9卷 861-865页

作者： Campos, PRA Onody, RN Univ Sao Paulo Inst Fis Sao Carlos Dept Fis Informat BR-13560970 Sao Carlos SP Brazil

Here we apply the wolff single-cluster algorithm to the site-bond-correlated Ising model and study its critical dynamical behavior. We have verified that the autocorrelation time diminishes in the presence of dilution and correlation, showing that the wolff algorithm performs even better in such situations. The critical dynamical exponents are also estimated.

关键词： wolff algorithm critical dynamical exponents Ising model

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