Binary trees, exploration processes, and an extented Ray--Knight Theorem

TL;DR

This paper explores the relationship between binary Galton--Watson trees and their exploration processes, extending classical results to large populations and deriving a generalized Ray--Knight theorem.

Contribution

It establishes a bijection between continuous-time binary Galton--Watson trees and exploration processes, and generalizes the Ray--Knight theorem for large populations.

Findings

Bijection between Galton--Watson trees and exploration processes

Limit theorems for renormalized quantities as population size grows

Generalization of the second Ray--Knight theorem

Abstract

We study the bijection between binary Galton--Watson trees in continuous time and their exploration process, both in the sub- and in the supercritical cases. We then take the limit over renormalized quantities, as the size of the population tends to infinity. We thus deduce Delmas' generalization of the second Ray--Knight theorem.