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Interacting Particle Systems

Thomas M. Liggett (Autor)

Buch | Hardcover

501 Seiten

1985
Springer-Verlag New York Inc.
978-0-387-96069-2 (ISBN)

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At what point in the development of a new field should a book be written about it? This question is seldom easy to answer. In the case of interacting particle systems, important progress continues to be made at a substantial pace. A number of problems which are nearly as old as the subject itself remain open, and new problem areas continue to arise and develop. Thus one might argue that the time is not yet ripe for a book on this subject. On the other hand, this field is now about fifteen years old. Many important of several basic models is problems have been solved and the analysis almost complete. The papers written on this subject number in the hundreds. It has become increasingly difficult for newcomers to master the proliferating literature, and for workers in allied areas to make effective use of it. Thus I have concluded that this is an appropriate time to pause and take stock of the progress made to date. It is my hope that this book will not only provide a useful account of much of this progress, but that it will also help stimulate the future vigorous development of this field.

I The Construction, and Other General Results.- 1. Markov Processes and Their Semigroups.- 2. Semigroups and Their Generators.- 3. The Construction of Generators for Particle Systems.- 4. Applications of the Construction.- 5. The Martingale Problem.- 6. The Martingale Problem for Particle Systems.- 7. Examples.- 8. Notes and References.- 9. Open Problems.- II Some Basic Tools.- 1. Coupling.- 2. Monotonicity and Positive Correlations.- 3. Duality.- 4. Relative Entropy.- 5. Reversibility.- 6. Recurrence and Transience of Reversible Markov Chains.- 7. Superpositions of Commuting Markov Chains.- 8. Perturbations of Random Walks.- 9. Notes and References.- III Spin Systems.- 1. Couplings for Spin Systems.- 2. Attractive Spin Systems.- 3. Attractive Nearest-Neighbor Spin Systems on Z1.- 4. Duality for Spin Systems.- 5. Applications of Duality.- 6. Additive Spin Systems and the Graphical Representation.- 7. Notes and References.- 8. Open Problems.- IV Stochastic Ising Models.- 1. Gibbs States.- 2. Reversibility of Stochastic Ising Models.- 3. Phase Transition.- 4. L2 Theory.- 5. Characterization of Invariant Measures.- 6. Notes and References.- 7. Open Problems.- V The Voter Model.- 1. Ergodic Theorems.- 2. Properties of the Invariant Measures.- 3. Clustering in One Dimension.- 4. The Finite System.- 5. Notes and References.- VI The Contact Process.- 1. The Critical Value.- 2. Convergence Theorems.- 3. Rates of Convergence.- 4. Higher Dimensions.- 5. Notes and References.- 6. Open Problems.- VII Nearest-Particle Systems.- 1. Reversible Finite Systems.- 2. General Finite Systems.- 3. Construction of Infinite Systems.- 4. Reversible Infinite Systems.- 5. General Infinite Systems.- 6. Notes and References.- 7. Open Problems.- VIII The Exclusion Process.- 1. Ergodic Theorems for Symmetric Systems.- 2. Coupling and Invariant Measures for General Systems.- 3. Ergodic Theorems for Translation Invariant Systems.- 4. The Tagged Particle Process.

Reihe/Serie	Grundlehren der Mathematischen Wissenschaften ; 276
Zusatzinfo	biography
Verlagsort	New York, NY
Sprache	englisch
Gewicht	875 g
Einbandart	gebunden
Themenwelt	Mathematik / Informatik ► Mathematik
ISBN-10	0-387-96069-4 / 0387960694
ISBN-13	978-0-387-96069-2 / 9780387960692
Zustand	Neuware