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Random Walks and Trees

(Lecture notes given at the 10th Symposium of Probability and Stochastic Processes and 1st France-Mexico Meeting of Probability and Stochastic Processes, CIMAT, Guanajuato, México, November 3-7, 2008.)

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-gout 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 (1999) and to the forthcoming book of Lyons and Peres, as well as to Duquesne and Le Gall (2002) and Le Gall (2005) 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.

Download: (pdf)
ESAIM: Proceedings 31 (2011) 1-39.

Or download my personal copy: (pdf)
THIS VERSION: June 21, 2010.