An introduction to Galton-Watson trees and their local limits

Abstract

The aim of this lecture is to give an overview of old and new resultson Bienaymé-Galton-Watson (BGW) trees. After introducing the framework of discretetrees, we first give alternative proofs of classical results on theextinction probability of BGW processes and on thedescription of the processes conditioned on extinction or onnon-extinction. Then, we study recent local limits of critical orsub-critical BGW trees conditioned to be large.

Figures (8)

• Figure 1: A finite tree ${\mathbf t}$.
• Figure 2: Two different planar trees.
• Figure 3: Generating function in the sub-critical (left), critical (middle) and super-critical (right) cases.
• Figure 4: In the super-critical case with $p(0)>0$ (and thus $0<q<1$): the generating functions $g_p$, $g_{\tilde{p}}$ in the lower sub-square and $g_{\hat{p}}$ in the upper sub-square (up to a scaling factor).
• Figure 5: Exemple of the third case in Equation \ref{['eq:inter-cf']} where $x_1\ne x_2$, ${\mathbf t}_1={\mathbf t}\circledast _{x_2}{{\mathbf t}'_1}$ (on the left), ${\mathbf t}_2={\mathbf t}\circledast _{x_1}{{\mathbf t}'_2}$ (in the middle) and with ${\mathbf t}_1\cup{\mathbf t}_2$ (on the right).

Theorems & Definitions (72)

• Lemma 2.1
• proof
• Definition 2.2: Branching property and BGW tree
• Lemma 2.3: BGW process
• Remark 2.4: Roots of $g(r)=r$
• Corollary 2.5: (Sub-)critical case
• Lemma 2.6: BGW tree conditioned on extinction
• proof
• Corollary 2.7: BGW process conditioned on extinction
• Lemma 2.8