Random Walks & Trees

Preface These notes provide an elementary and self-contained introduction to branching random walks. Chapter 1 gives a brief overview of Galton–Watson trees, whereas Chapter 2 presents the classical law of large numbers for branching random walks. These two short chapters are not exactly indispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-goût to the main part, Chapter 3, where branching random walks are studied from a deeper point of view, and are connected to the model of directed polymers on a tree. Tree-related random processes form a rich and exciting research subject. These notes cover only special topics. For a general account, we refer to the St-Flour lecture notes of Peres [44] and to the forthcoming book of Lyons and Peres [39], as well as to Duquesne and Le Gall [22] and Le Gall [34] for continuous random trees. I am grateful to the organizers of the Symposium for the kind invitation, to the participants for the very enjoyable week, and to my co-authors for sharing the pleasure of random climbs.