Preface

Summary

The title of this book is owed in large part to my personal motivation to study the material I present here. It is rooted in the problem of so-called mean-field models of spin glasses. I will not go into a discussion of the physical background of these systems (see, e.g., [25]). The key mathematical objects associated with them are random functions (called Hamiltonians on some highdimensional space, e.g. ﹛−1, 1﹜ⁿ. The standard model here is the Sherrington– Kirkpatrick model, introduced in a seminal paper [103] in 1972. Here the Hamiltonian can be seen as a Gaussian process indexed by the hypercube ﹛−1, 1﹜ⁿ whose covariance is a function of the Hamming distance. The attempt to understand the structure of these objects has given rise to the remarkable heuristic theory of replica symmetry breaking developed by Parisi and collaborators (see the book [91]). A rigorous mathematical corroboration of this theory was obtained only rather recently through the work of Talagrand [109, 108, 110, 111], Guerra [63], Aizenman, Simms and Starr [6] and Panchenko [97], to name the most important ones.

A second class of models that are significantly more approachable by rigorous mathematics was introduced by Derrida and Gardner [46, 59]. Here the Hamming distance was replaced by the lexicographic ultra-metric on ﹛−1, 1﹜. The resulting class of models are called the generalised random energy models (GREM). These processes can be realised as branching random walks with Gaussian increments and thus provide the link to the general topic of this book.

From branching random walks it is a small step to branching Brownian motion (BBM), a classical object of probability theory, introduced by Moyal [1, 92] in 1962. BBM has been studied over the last 50 years as a subject of interest in its own right, with seminal contributions by McKean [90], Bramson [33, 34], Lalley and Sellke [83], Chauvin and Rouault [39, 40] and others. Recently, the field has experienced a revival with many remarkable contributions and repercussions in other areas.