Mean field models for spin glasses, Volume II, advanced replica-symmetry and low temperature / Michel Talagrand

Contient des exercices

Bibliogr. p. 623-627. Index

The first part of this volume, "Advanced replica-symmetry'', is a continuation, at a deeper level, of the analysis of the four models introduced in Volume I: Shcherbina-Tirozzi model, perceptron, Hopfield model, Sherrington-Kirkpatrick (SK) model. The gain in accuracy is quite impressive in Chapter 10, where new subtle ideas and fine arguments are added.
The second part of this volume, "Low temperature'', deals with replica-symmetry breaking. An ultimate goal is a proof of the celebrated formula of G. Parisi, which gives the value of the limiting free energy of the SK model. The formula, derived by Parisi in 1980 using a non-rigorous replica method, is a true mathematical challenge; it was proved by the author in 2006 [Ann. of Math. (2) 163 (2006), no. 1, 221–263; MR2195134]. In Chapter 13 the main ideas are explained in the simplest case. The heart of the matter is the extended Chapter 14, where serious technicalities come after presenting the central line of the proof. Chapter 15 can be read without mastering all these technicalities. It covers many fascinating features of the Parisi solution, including ultrametricity, Ghirlanda-Guerra identities, Poisson-Dirichlet cascades, and random overlap structures. The volume ends with the p-spin model.
Not only does this book lead the reader towards the proof of the Parisi formula, but it also reveals the beautiful structure which is embedded in the model and the "new territories'' mentioned in the dedication. With its depth and profusion of important techniques, this book, together with Volume I, opens new perspectives in probability theory and statistical mechanics of disordered systems. It will be a source of inspiration for generations of mathematicians. (MSN)