Interacting particle systems.

*(English)*Zbl 0559.60078
Grundlehren der Mathematischen Wissenschaften, 276. New York etc.: Springer-Verlag. XV, 488 p. DM 196.00 (1985).

Interacting particle systems serve as models for as different things as ferromagnetism, spread of infection or turbulences in liquids. The most interesting phenomena in such systems are the multiplicity of existing phases (various directions or strength of magnetisation) and the existence of critical values for certain parameters where spontaneous changes in the spectrum of possible phases occur (Curie temperature).

Mathematically such a system is a Markov process. The above phenomena rewrite in terms of invariant measures and ergodicity. Thus the first chapter of the book is about general results on Markov processes. The relation to semi-groups and some theorems on existence and uniqueness of invariant measures are stated. Markov processes for spin systems are constructed from a collection of transition measures. Then the martingale approach to Markov processes is discussed. Mutual singularity of measures corresponding to different epochs makes the difference between interacting particle systems and other more common Markov processes. General results being very rare, the theory is an account of tools for an investigation of special systems. These tools are well-known from other fields of probability but are often used here in a different manner.

Coupling, duality, relative entropy, and reversibility are such tools introduced in chapter II. A new result on the stability of positive recurrence for Markov chains without imposing assumptions on the moments is presented. In chapter III these tools are applied to general spin systems. The analysis of the most important spin systems (Ising-model, voter-model, contact process, exclusion process, and nearest particle process) is almost complete. They are the simplest examples where the above phenomena occur. These systems are treated rather independently in chapters IV through VIII. The concept of potentials and Gibbs states is developed. Chapter IX is about linear systems with state space \([0,\infty)^ S.\)

Which subjects are not included? These are infinite systems of stochastic differential equations, measure-valued diffusions, shape theory, renormalization theory, and some well known models as the classical Heisenberg model or Dyson’s hierarchical models. Also discrete time systems are not mentioned.

”Notes and references” and ”open problems” at the end of each chapter give a good insight into the state of the art. That is helpful for newcomers to master the great number of research papers.

Mathematically such a system is a Markov process. The above phenomena rewrite in terms of invariant measures and ergodicity. Thus the first chapter of the book is about general results on Markov processes. The relation to semi-groups and some theorems on existence and uniqueness of invariant measures are stated. Markov processes for spin systems are constructed from a collection of transition measures. Then the martingale approach to Markov processes is discussed. Mutual singularity of measures corresponding to different epochs makes the difference between interacting particle systems and other more common Markov processes. General results being very rare, the theory is an account of tools for an investigation of special systems. These tools are well-known from other fields of probability but are often used here in a different manner.

Coupling, duality, relative entropy, and reversibility are such tools introduced in chapter II. A new result on the stability of positive recurrence for Markov chains without imposing assumptions on the moments is presented. In chapter III these tools are applied to general spin systems. The analysis of the most important spin systems (Ising-model, voter-model, contact process, exclusion process, and nearest particle process) is almost complete. They are the simplest examples where the above phenomena occur. These systems are treated rather independently in chapters IV through VIII. The concept of potentials and Gibbs states is developed. Chapter IX is about linear systems with state space \([0,\infty)^ S.\)

Which subjects are not included? These are infinite systems of stochastic differential equations, measure-valued diffusions, shape theory, renormalization theory, and some well known models as the classical Heisenberg model or Dyson’s hierarchical models. Also discrete time systems are not mentioned.

”Notes and references” and ”open problems” at the end of each chapter give a good insight into the state of the art. That is helpful for newcomers to master the great number of research papers.

Reviewer: T.Fritz

##### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60Jxx | Markov processes |