Power of the Crowd

The sum of powers of subtree sizes for conditioned Galton-Watson trees

We study the additive functional $X_n(α)$ on conditioned Galton-Watson trees given, for arbitrary complex $α$, by summing the $α$th power of all subtree sizes. Allowing complex $α$ is advantageous, even for the study of real $α$, since it allows us to use powerful results from the theory of analytic functions in the proofs. For $\Reα< 0$, we prove that $X_n(α)$, suitably normalized, has a complex normal limiting distribution; moreover, as processes in $α$, the weak convergence holds in the space of analytic functions in the left half-plane. We establish, and prove similar process-convergence e…

Invariance and attraction properties of Galton-Watson trees

We give a description of invariants and attractors of the critical and subcritical Galton-Watson tree measures under the operation of Horton pruning (cutting tree leaves with subsequent series reduction). Under a regularity condition, the class of invariant measures consists of the critical binary Galton-Watson tree and a one-parameter family of critical Galton-Watson trees with offspring distribution $\{q_k\}$ that has a power tail $q_k\sim Ck^{-(1+1/q_0)}$, where $q_0\in(1/2,1)$. Each invariant measure has a non-empty domain of attraction under consecutive Horton pruning, specified by the ta…

Convergence of the height process of supercritical Galton-Watson forests with an application to the configuration model in the critical window

We show joint convergence of the Lukasiewicz path and height process for slightly supercritical Galton-Watson forests. This shows that the height processes for supercritical continuous state branching processes as constructed by Lambert (2002) are the limit under rescaling of their discrete counterparts. Unlike for (sub-)critical Galton-Watson forests, the height process does not encode the entire metric structure of a supercritical Galton-Watson forest. We demonstrate that this result is nonetheless useful, by applying it to the configuration model with an i.i.d. power-law degree sequence in …

Conditioned Galton-Watson trees: The shape functional, and more on the sum of powers of subtree sizes and its mean

For a complex number $α$, we consider the sum of the $α$th powers of subtree sizes in Galton--Watson trees conditioned to be of size $n$. Limiting distributions of this functional $X_n(α)$ have been determined for $\Reα\neq 0$, revealing a transition between a complex normal limiting distribution for $\Reα< 0$ and a non-normal limiting distribution for $\Reα> 0$. In this paper, we complete the picture by proving a normal limiting distribution, along with moment convergence, in the missing case $\Reα= 0$. The same results are also established in the case of the so-called shape functional $X_n'(…

Galton-Watson过程、简单树种及其Khintchin族

Galton-Watson processes, simple varieties of trees and Khinchin families

本文通过拉格朗日逆公式建立了树种、Galton-Watson过程和Khintchin族之间的联系，并提出了一种基于幂级数系数的高效模拟方法。

Ergodic property for Galton-Watson processes in which individuals have variable life times

This paper is concerned with an extended Galton-Watson process so as to allow individuals to live and reproduce for more than one unit time. We assume that each individual can live $k$ seasons (time-units) with probability $h_k$, and produce $m$ offspring with probability $p_m$ during each season. These can be seen as Galton-Watson processes with countably infinitely many types in which particles of type $i$ may only have offspring of type $i+1$ and type $1$. Let $\textbf{M}$ be its mean progeny matrix and $γ$ be the convergence radius of the power series $\sum_{k\geq 0}r^k(\textbf{M}^k)_{ij}$…