An introduction to Galton-Watson trees and their local limits

Lecture given in Hammamet, December 2014.The main object of this course given in Hammamet (December 2014) is the so-called Galton-Watson process.We introduce in the first chapter of this course the framework of discrete random trees. We then use this framework to construct GW trees that describe the genealogy of a GW process. It is very easy to recover the GW process from theGW tree as it is just the number of individuals at each generation. We then give alternativeproofs of classical results on GW processes using the tree formalism. We focus in particular onthe extinction probability (which was the first question of F. Galton) and on the description ofthe processes conditioned on extinction or non extinction.In a second chapter, we focus on local limits of conditioned GW trees. In the critical andsub-critical cases, the population becomesa.s. extinct and the associated genealogical tree is finite. However, it has a small but positiveprobability of being large (this notion must be precised). The question that arises is to describethe law of the tree conditioned of being large, and to say what exceptional event has occurredso that the tree is not typical. A first answer to this question is due to H. Kesten whoconditioned a GW tree to reach height n and look at the limit in distribution when n tends toinfinity. There are however other ways of conditioning a tree to be large: conditioning on havingmany vertices, or many leaves... We present here very recent general results concerning this kindof problems due to the authors of this course and completed by results of X. He