Gaussian processes on trees. From spin glasses to branching Brownian motion (Q2825424)

This book covers a wide range of topics. Among them are, e.g., extreme values of random sequences, point processes, spin glasses, branching Brownian motion (BBM), PDEs related to BBMs, extremal processes and their relations with BBMs.NEWLINENEWLINEAs the author explains, he was interested in the so-called mean-field models of spin glasses. The Hamiltonians involved can be considered as Gaussian processes indexed by an \(n\)-dimensional hypercube with covariance functions expressed in terms of Hamming distance. This justifies why such a title is chosen.NEWLINENEWLINEIn order to understand the material, the reader definitely has to possess a strong background in analysis and probability.NEWLINENEWLINEChapters 1--4 provide the reader with the necessary knowledge on topics such as extremal sequences and processes, point processes, and spin glasses. Not only important results are included, but they are given with enough details of their proofs. This variety of ideas and techniques are interesting by themselves, but they all are an excellent basis to move to the main object in the book, namely, the analysis of BBMs. This is done in Chapter 5 and the next ones. It is interesting to mention that the (Fisher)-Kolmogorov-Petrovsky-Piskunov PDEs (F-KPP) play a fundamental role due to the simple reason that the solutions of such equations are expressed as functionals of a BBM. The formulas are of Kac-Feynman type. The author discusses some beautiful and deep connections between a fine behaviour of the solutions of F-KPP equations and fine properties of BBM.NEWLINENEWLINEWhen discussing most of the questions, the author pays good attention to both, ideas and techniques. He presents a large number of results, many of them are non-trivial limit theorems. Some results are classical in the field, others are quite new, published very recently. While some of the results belong to the author, credit is given to several other contributors in the area.NEWLINENEWLINEBesides the many results given with their proofs, the author includes useful bibliographical notes in the end of each chapter. The book ends with a comprehensive list of 117 references and Index.NEWLINENEWLINEThis is a well-written book on hot topics from modern stochastics and its applications. The book can be recommended to researchers and university graduate students.