# zbMATH — the first resource for mathematics

Phase transitions of interacting particle systems. (English) Zbl 0847.60088
Singapore: World Scientific. xv, 228 p. (1994).
The phenomenon of phase transitions is a challenge subject not only for mathematicians but also for the scientists in various different fields. In the past three – four decades, a number of mathematical tools has been presented: the Peierls method, the Pirogov-Sinai theory, the reflection positivity, the percolation approach, the renormalization approach and so on. The book emphasizes the story of phase transitions for interacting particle systems. The main aim is to estimate the critical values and the order parameters, their values are still unknown for the models treated in the book. The models are carefully chosen and typical: the basic contact process, the $$\theta$$-contact process, the diffusive contact processes, the long-range contact processes and the uniform nearest particle systems. From the book, the readers have a chance to enjoy several different approaches and their comparison: Harris-FKG inequality, Holley-Liggett’s method, Ziezold-Grillenberger’s method, Griffeath’s method and the one by the author with M. Katori. Roughly speaking, the methods are based on the correlation functions. A large number of correlation identities or inequalities is either proved or conjectured in the book or in the author’s subsequent lecture notes given in Universidade de São Paulo (Brazil) in the Spring of 1996. The book is well organized and friendly written. It should be helpful for the readers from various fields who are interested in the fascinated subject.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60-02 Research exposition (monographs, survey articles) pertaining to probability theory 82-02 Research exposition (monographs, survey articles) pertaining to statistical mechanics