The random-cluster model

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The random-cluster model / Geoffrey Grimmett

資料種別:: 図書
出版情報:: Berlin : Springer, c2006
形態:: xiii, 377 p. ; 24 cm
シリーズ名:: Die Grundlehren der mathematischen Wissenschaften ; 333 <BA00007422>
著者名:: Grimmett, Geoffrey <DA01920467>
ISBN:: 9783540328902 [3540328904]
書誌ID:: BA77378096

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The random-cluster model. 2nd corr. print.
Geoffrey Grimmett , Springer , 2009

書誌詳細

注記:

Includes bibliographical references (p. [353]-372) and index

主題:

Ferromagnetism; Phase transformations (Statistical physics); Stochastic processes

言語:

英語

目次情報:

Random-Cluster Measures / 1：

Introduction / 1.1：

Random-cluster model / 1.2：

Ising and Potts models / 1.3：

Random-cluster and Ising/Potts models coupled / 1.4：

The limit as q [down arrow] 0 / 1.5：

Basic notation / 1.6：

Monotonic Measures / 2：

Stochastic ordering of measures / 2.1：

Positive association / 2.2：

Influence for monotonic measures / 2.3：

Sharp thresholds for increasing events / 2.4：

Exponential steepness / 2.5：

Fundamental Properties / 3：

Conditional probabilities / 3.1：

Differential formulae and sharp thresholds / 3.2：

Comparison inequalities / 3.4：

Partition functions / 3.5：

Domination by the Ising model / 3.7：

Series and parallel laws / 3.8：

Negative association / 3.9：

Infinite-Volume Measures / 4：

Infinite graphs / 4.1：

Boundary conditions / 4.2：

Infinite-volume weak limits / 4.3：

Infinite-volume random-cluster measures / 4.4：

Uniqueness via convexity of pressure / 4.5：

Potts and random-cluster models on infinite graphs / 4.6：

Phase Transition / 5：

The critical point / 5.1：

Percolation probabilities / 5.2：

Uniqueness of random-cluster measures / 5.3：

The subcritical phase / 5.4：

Exponential decay of radius / 5.5：

Exponential decay of volume / 5.6：

The supercritical phase and the Wulff crystal / 5.7：

Uniqueness when q < 1 / 5.8：

In Two Dimensions / 6：

Planar duality / 6.1：

The value of the critical point / 6.2：

First-order phase transition / 6.3：

General lattices in two dimensions / 6.5：

Square, triangular, and hexagonal lattices / 6.6：

Stochastic Lowner evolutions / 6.7：

Duality in Higher Dimensions / 7：

Surfaces and plaquettes / 7.1：

Basic properties of surfaces / 7.2：

A contour representation / 7.3：

Polymer models / 7.4：

Discontinuous phase transition for large q / 7.5：

Dobrushin interfaces / 7.6：

Probabilistic and geometric preliminaries / 7.7：

The law of the interface / 7.8：

Geometry of interfaces / 7.9：

Exponential bounds for group probabilities / 7.10：

Localization of interface / 7.11：

Dynamics of Random-Cluster Models / 8：

Time-evolution of the random-cluster model / 8.1：

Glauber dynamics / 8.2：

Gibbs sampler / 8.3：

Coupling from the past / 8.4：

Swendsen-Wang dynamics / 8.5：

Coupled dynamics on a finite graph / 8.6：

Box dynamics with boundary conditions / 8.7：

Coupled dynamics on the infinite lattice / 8.8：

Simultaneous uniqueness / 8.9：

Flows in Poisson Graphs / 9：

Potts models and flows / 9.1：

Flows in the Ising model / 9.2：

Exponential decay for the Ising model / 9.3：

The Ising model in four and more dimensions / 9.4：

On Other Graphs / 10：

Mean-field theory / 10.1：

On complete graphs / 10.2：

Main results for the complete graph / 10.3：

The fundamental proposition / 10.4：

The size of the largest component / 10.5：

Proofs of main results for complete graphs / 10.6：

The nature of the singularity / 10.7：

Large deviations / 10.8：

On a tree / 10.9：

The critical point for a tree / 10.10：

(Non-)uniqueness of measures on trees / 10.11：

On non-amenable graphs / 10.12：

Graphical Methods for Spin Systems / 11：

Random-cluster representations / 11.1：

The Potts model / 11.2：

The Ashkin-Teller model / 11.3：

The disordered Potts ferromagnet / 11.4：

The Edwards-Anderson spin-glass model / 11.5：

The Widom-Rowlinson lattice gas / 11.6：

The Origins of FK(G) / Appendix：

List of Notation

References

Index

Random-Cluster Measures / 1：

Introduction / 1.1：

Random-cluster model / 1.2：

Ising and Potts models / 1.3：

Random-cluster and Ising/Potts models coupled / 1.4：

The limit as q [down arrow] 0 / 1.5：

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