Introduction to the replica theory of disordered statistical systems (Q2706562)

Disordered systems, and in particular spin glasses, have become a most fascinating topic that is gaining increasing attention from a mathematical audience. One of the reasons for this fascination is the fact that theoretical physics has devised a heuristic analytic approach to the problems in this field, known as the replica method, that allows to solve problems related to high-dimensional random fields that none of the mathematical tools that are currently available can solve. In this situation, a new book that promises an introduction to the replica theory of disordered systems is a priori very welcome. There are, in fact, quite a number of physics textbooks on the subject, including the by now classic ``Spin glass theory and beyond'' by \textit{M. Mézard, G. Parisi}, and \textit{M. A. Virasoro} (1987; Zbl 0992.82500), but none of these is very easy to access for the more mathematically minded. A modern text that would help bridge the gap between the two cultures would thus be most useful. NEWLINENEWLINENEWLINEUnfortunately, the text under review will not serve this purpose. To start, all of the recent mathematical progress is ignored. That is a pity, since some of the results obtained by Talagrand (see, e.g., Zbl 0999.82045) and others justify, if only in the simplest cases, the predictions of the replica method, which, as the author points out ``may not be so crazy after all''. If these results are maybe too recent to have been perceived by the author, another omission concerns the fundamental contributions to the analysis of the phase transition in the random field Ising model by \textit{J. Bricmont} and \textit{A. Kupiainen} [Phase transition in the 3d random field Ising model. Commun. Math. Phys. 116, 539-572 (1988)] and \textit{M. Aizenman} and \textit{J. Wehr} [Commun. Math. Phys. 130, 489-528 (1990; Zbl 0714.60090)]. These works are more than a decade old and represent milestone achievements of mathematical physics on questions that were under dispute since different sophisticated heuristic arguments gave diverging answers on the fundamental question on the lower critical dimension in this model. These developments do not seem to concern the author, even though he discusses the random field Ising model extensively. Of equally little concern are the intrinsic mathematical difficulties of the replica method, which are by and large kept under the carpet. The reader is expected to accept this method for its merits, a possible, but maybe at the present time not the only way to advertise the subject. NEWLINENEWLINENEWLINEThese shortcomings put aside, the text provides an introduction to the replica method and its various areas of applications. The first part of the book covers spin glasses, the second the effects of randomness on critical exponents (largely in the context of random bond Ising models), and the last part covers other systems: the random field Ising model and directed polymers in random potentials. The breadth of the exposition, and in particular the fact that also non-mean field systems are covered, is the main merit of this text and illustrates the wide range of applicability of this method. For this reason, the book will be a useful reference for the researcher and a reasonable first introduction to the field for graduate students entering the field.