Random Walks and Trees

Abstract

These notes provide an elementary and self-contained introduction to branching ran- dom walks. Section 1 gives a brief overview of Galton-Watson trees, whereas Section 2 presents the classical law of large numbers for branching random walks. These two short sections are not exactly in- dispensable, but they introduce the idea of using size-biased trees, thus giving motivations and an avant-gout to the main part, Section 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 (47) and to the forthcoming book of Lyons and Peres (42), as well as to Duquesne and Le Gall (23) and Le Gall (37) for continuous random trees.