Book review

This monograph presents an important topic of recent research well characterized by its title. It gives a full report on the interplay of three originally separated fields of mathematical and theoretical physics, viz. non-Markovian random walks as part of the theory of stochastic processes, critical phenomena accompanying phase transitions in many-particle systems as part of statistical physics, and triviality, i.e. absence of genuine interactions, as part of nonlinear quantum field theory.

Random walks and Brownian motions are ubiquitous in physics. In particular, their intersection properties are nontrivial and have been studied for decades. Now they play an important role in the theory of critical phenomena and in quantum field theory. A quantum field can indeed always be described as a gas of random walks, the intersections of which represent precisely the interactions in the nonlinear field. This fact is the key for understanding the new developments starting in 1980.

The volume is divided into three parts with altogether fifteen chapters. The first part gives a detailed survey of critical phenomena, quantum field theory, random walks and random surfaces. The problems considered include relativistic quantum field theory and Euclidean field theory, phase transitions in classical spin systems, scale transformations and the renormalization group, random walks as Euclidean field theory, and various random surface models.

The second part on random-walk models and random-walk representations of classical lattice spin systems embraces random-walk models without and with a magnetic field and a discussion of the long row of the so-called correlation inequalities.

The third part on consequences for critical phenomena and quantum field theory contains a thorough discussion and proofs of a lot of inequalities for critical exponents including a review of the Landau-Ginzburg theory of phase transitions, hyperscaling inequalities and extrapolation principles; finally there follows the treatment of continuum limits of lattice theories leading to the problem of triviality of quantum fields.

The text is supplemented by a valuable list (26 pp.) of 527 references in alphabetical order with titles and a detailed subject index (14 pp.). There are lists of tables and of figures.

This book is to be classified as a unique research monograph on a high professional level. It is not written in the abstract style of definition, theorem and proof, but gives full attention to physical motivations, the relevance of concepts, and overlapping related matters. Thus the text is well readable providing a huge amount of scientific information. Nevertheless, the book is not exhaustive, e.g. the random-walk representations of superrenormalizable field theories in space-time dimensions d < 4 are only sketched, and percolation theory, branched polymers, and conformal invariance in quantum field theory and statistical physics are outside the scope of the work.

The volume is well produced, and the printing is excellent. The well-known authors have written a distinguished book which can be warmly recommended to all research workers, teachers, and graduate students of mathematical and theoretical physics. For libraries it is certainly a must.