Jan M. Swart

The culmination of years of teaching experience, this book provides a modern introduction to the mathematical theory of interacting particle systems. Assuming a background in probability and measure theory, it has been designed to support a one-semester course at a Master or Ph.D. level. It also provides a useful reference for researchers, containing several results that have not appeared in print in this form before. An emphasis is placed on graphical representations, which are used to give a construction that is intuitively easier to grasp than the traditional generator approach. Also included is an extensive look at duality theory, along with discussions of mean-field methods, phase transitions and critical behaviour. The text is illustrated with the results of numerical simulations and features exercises in every chapter. The theory is demonstrated on a range of models, reflecting the modern state of the subject and highlighting the scope of possible applications.

It is known that certain one-dimensional nearest-neighbour random walks in i.i.d. random space-time environments have diffusive scaling limits. Here, in the continuum limit, the random environment is represented by a â??stochastic flow of kernels', which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment.