Probability on graphs: random processes on graphs and lattices, by Geoffrey Grim-

Growth of the field of mathematics called Interacting Particle Systems (IPS) was, to a great extent, catalyzed by Liggett’s 1985 monograph [12]. Thirty years later this IPS field (together with the closely related percolation field) remains an active area within mathematical probability and the theorem-proof end of statistical physics, with its MSC primary classification 60K35 containing about 170 papers a year. In brief, IPS studies models over a graph: each vertex (or site) is in some state; vertices interact with neighboring vertices at random intervals and update their states according to some rule. For instance in the contact process, a natural toy model for epidemics, a vertex is either infected or healthy: healthy sites become infected at rate λ times the number of infected neighbors, and infected sites become healthy at constant rate one (rate means probability per unit time). Liggett’s book gave a masterful account of work on several models over the previous 15 years, for which a major original motivation had been rigorous study of the statistical physics notion of phase transitions, formalized as qualitative changes in behavior of a process defined on the infinite d-dimensional lattice as parameters pass through a critical value. For instance in the contact process, the epidemic will die out if λ is smaller than the critical value, but has a chance to survive forever if λ is larger. Not only has IPS remained an active field within statistical physics and mathematical probability, but similar models have been used in a surprising range of other quantitative disciplines. To cite just two recent graduate texts, [14] treats models relevant to topics within computational complexity (satisfiability problems) and information theory (error-correcting codes); and [7] shows how continuum analogs of percolation models are fundamental to mathematical study of wireless network communication. A more familiar context is provided by social networks. Whatever you think a social network does, it involves some notion of exchange of information, and toy mathematical models are inevitably somewhat similar to the models from IPS (for instance there are many variants of the voter model for how your opinions are affected by your friends’ opinions). See [3] for a wide-ranging overview of such literature, and [1] for novel toy models in the spirit of IPS. In all these broader disciplines the appropriate underlying “geometry” (described by the graph) is context-dependent, not the 3-dimensional lattice used classically in physics models for phenomena such as ferromagnetism. Writing a subsequent book which resembled Liggett’s in being both introductory and comprehensive seems quite impractical, not only because of the size of the field but also because it would need to duplicate much of Liggett’s material. So the subsequent book literature consists of a small set of lecture notes and specialized monographs, such as [4, 6, 8, 9, 13]. This makes it unfortunately difficult for a newcomer to know how to get started in the field—how to start learning the techniques, models and results behind today’s research frontier. The book under review serves