# Condensation phenomena in random trees - Lecture 1

Date: Tue, Jul 23, 2024

Location: CRM, Montreal

Conference: 2024 CRM-PIMS Summer School in Probability

Subject: Mathematics, Probability

Class: Scientific

### Abstract:

Consider a population that undergoes asexual and homogeneous reproduction over time, originating from a single individual and eventually ceasing to exist after producing a total of n individuals. What is the order of magnitude of the maximum number of children of an individual in this population when n tends to infinity? This question is equivalent to studying the largest degree of a large Bienaymé-Galton-Watson random tree. We identify a regime where a condensation phenomenon occurs, in which the second greatest degree is negligible compared to the greatest degree. The use of the "one-big jump principle" of certain random walks is a key tool for studying this phenomenon. Finally, we discuss applications of these results to other combinatorial models.