Interacting particle systems at Saint-Flour. Reprint of lectures originally published in the Lecture Notes in Mathematics volumes 1608 (1995), 1464 (1991), 589 (1977), and 390 (1974).

*(English)*Zbl 1248.82021
Probability at Saint-Flour. Berlin: Springer (ISBN 978-3-642-25297-6/pbk). viii, 331 p. (2012).

This book is a reprint of lectures that were originally published in four volumes of the Lecture Notes in Mathematics. They are devoted to considerations about interacting particle systems at Saint-Flour Probability Summer School. It is organized every year by the Université Blaise Pascal at Clermont-Ferrand in France. The considered book contains a presentation of four lectures that are as follows.

Lecture 1 is an introduction to Markov processes with parameter \(Z_v\) – written in French. It consists of seven chapters, namely, (a) random fields and thermodynamic limits; (b) Markov states and Gibbs finished products; (c) ballasts of Markov states and Gibbs \(Z_v\); d) phase transition for the Ising model of gas; (e) variational characterization of Gibbs states; (f) temporal evolutions; (g) Gaussian Markov fields.

Lecture 2 is on stochastic evolution on infinite systems of interacting particles. It has two major parts. Part I (Spin-flip processes) contains (1) existence results and first ergodic theorems; (2) usage of coupling and duality techniques. Part II (exclusion processes) contains (1) existence results and identification of simple invariant measures; (2) symmetric and asymmetric simple exclusion processes; (3) exclusion process with speed change.

Lecture 3 (topics in propagation of chaos) is comprised of generalizations and basic examples, local interactions leading to Burgers’ equation, the constant mean free travel time regime and the uniqueness of the Boltzmann process.

In the last lecture presented in this book, a set of reading devoted to considerations on particle systems with special attention paid on percolation substructures, threshold models, cyclic models, long-range limits and predator prey systems is given.

Lecture 1 is an introduction to Markov processes with parameter \(Z_v\) – written in French. It consists of seven chapters, namely, (a) random fields and thermodynamic limits; (b) Markov states and Gibbs finished products; (c) ballasts of Markov states and Gibbs \(Z_v\); d) phase transition for the Ising model of gas; (e) variational characterization of Gibbs states; (f) temporal evolutions; (g) Gaussian Markov fields.

Lecture 2 is on stochastic evolution on infinite systems of interacting particles. It has two major parts. Part I (Spin-flip processes) contains (1) existence results and first ergodic theorems; (2) usage of coupling and duality techniques. Part II (exclusion processes) contains (1) existence results and identification of simple invariant measures; (2) symmetric and asymmetric simple exclusion processes; (3) exclusion process with speed change.

Lecture 3 (topics in propagation of chaos) is comprised of generalizations and basic examples, local interactions leading to Burgers’ equation, the constant mean free travel time regime and the uniqueness of the Boltzmann process.

In the last lecture presented in this book, a set of reading devoted to considerations on particle systems with special attention paid on percolation substructures, threshold models, cyclic models, long-range limits and predator prey systems is given.

Reviewer: Dominik Strzałka (Rzeszów)

##### MSC:

82B30 | Statistical thermodynamics |

82C22 | Interacting particle systems in time-dependent statistical mechanics |

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

82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |

82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |

82C35 | Irreversible thermodynamics, including Onsager-Machlup theory |