Stochastic Interacting Systems: Contact, Voter and Exclusion ProcessesInteractive Particle Systems is a branch of Probability Theory with close connections to Mathematical Physics and Mathematical Biology. In 1985, the author wrote a book (T. Liggett, Interacting Particle System, ISBN 3-540-96069) that treated the subject as it was at that time. The present book takes three of the most important models in the area, and traces advances in our understanding of them since 1985. In so doing, many of the most useful techniques in the field are explained and developed, so that they can be applied to other models and in other contexts. Extensive Notes and References sections discuss other work on these and related models. Readers are expected to be familiar with analysis and probability at the graduate level, but it is not assumed that they have mastered the material in the 1985 book. This book is intended for graduate students and researchers in Probability Theory, and in related areas of Mathematics, Biology and Physics. |
Contents
| 1 | |
| 11 | |
| 21 | |
The Martingale | 29 |
The Process on the Integer Lattice | 45 |
The Process on 1 Nd | 71 |
The Process on the Homogeneous Tree Ta | 78 |
29 | 88 |
Notes and References | 201 |
Exclusion Processes | 209 |
The Process X Identifies the Shock | 219 |
Behavior of the Shock First Moments | 238 |
Central Limit Behavior of the Shock | 253 |
Invariant Measures for Processes on | 261 |
The Partition Function | 269 |
An Application the Process with a Blockage | 276 |
44 | 94 |
Notes and References | 125 |
Voter Models | 139 |
Models with General Threshold and Range | 146 |
Notes and References | 298 |
Bibliography | 317 |
Index 331 | 330 |
Other editions - View all
Stochastic Interacting Systems: Contact, Voter and Exclusion Processes Thomas M. Liggett Limited preview - 1999 |
Stochastic Interacting Systems: Contact, Voter and Exclusion Processes Thomas M. Liggett No preview available - 2014 |
Stochastic Interacting Systems: Contact, Voter and Exclusion Processes Thomas M. Liggett No preview available - 2010 |
Common terms and phrases
A₁ active path analogous argument asymptotic B₁ bounded branching process branching random walk C₁ central limit theorem compute consider constant multiple construction convergence corresponding coupling critical value defined dimensional duality Durrett equation ergodic exclusion process exponential exponential decay fact finite function gives graphical representation implies independent inequality infection arrows initial configuration initial distribution integer Lemma Liggett Markov chain Markov property martingale monotonicity N₁ n²¹³ nearest neighbor nonnegative nontrivial invariant measure Note oriented percolation P(en parameter particle systems pivotal interval Poisson processes positive probability measure process on Zd product measure proof of Theorem Proposition proved random variables Recall result right side satisfies Schonmann second class particle Section sequence statement stationary Suppose survive strongly symmetric Theorem B21 threshold contact process threshold voter model transition translation invariant X₁ zero