Limit theorems for conditioned non-generic Galton-Watson trees

TL;DR

This paper investigates the structure and scaling limits of conditioned non-generic Galton-Watson trees, revealing that their height grows logarithmically and they exhibit condensation phenomena, contrasting with critical cases.

Contribution

The study provides a detailed analysis of non-generic Galton-Watson trees, highlighting their unique condensation behavior and logarithmic height growth, which differs from known critical case results.

Findings

Height grows logarithmically with tree size

Condensation results in a unique high-degree vertex

Fluctuations around the condensation vertex are characterized

Abstract

We study a particular type of subcritical Galton--Watson trees, which are called non-generic trees in the physics community. In contrast with the critical or supercritical case, it is known that condensation appears in certain large conditioned non-generic trees, meaning that with high probability there exists a unique vertex with macroscopic degree comparable to the total size of the tree. Using recent results concerning subexponential distributions, we investigate this phenomenon by studying scaling limits of such trees and show that the situation is completely different from the critical case. In particular, the height of such trees grows logarithmically in their size. We also study fluctuations around the condensation vertex.