Invariance principles for Galton-Watson trees conditioned on the number of leaves

TL;DR

This paper studies the asymptotic structure of critical Galton-Watson trees conditioned on having many leaves, revealing convergence to stable Lévy process excursions and analyzing maximum degree distribution.

Contribution

It establishes the convergence of rescaled tree encodings to stable Lévy process excursions and extends prior results to trees conditioned on large degree sets.

Findings

Rescaled Lukasiewicz path converges to a stable Lévy excursion.

Rescaled height function converges to a continuous-time height process.

Distribution of maximum degree in large-leaf trees analyzed.

Abstract

We are interested in the asymptotic behavior of critical Galton-Watson trees whose offspring distribution may have infinite variance, which are conditioned on having a large fixed number of leaves. We first find an asymptotic estimate for the probability of a Galton-Watson tree having $n$ leaves. Secondly, we let $t_n$ be a critical Galton-Watson tree whose offspring distribution is in the domain of attraction of a stable law, and conditioned on having exactly $n$ leaves. We show that the rescaled Lukasiewicz path and contour function of $t_n$ converge respectively to $X^{exc}$ and $H^{exc}$, where $X^{exc}$ is the normalized excursion of a strictly stable spectrally positive L\'evy process and $H^{exc}$ is its associated continuous-time height function. As an application, we investigate the distribution of the maximum degree in a critical Galton-Watson tree conditioned on having a large number of leaves. We also explain how these results can be generalized to the case of Galton-Watson trees which are conditioned on having a large fixed number of vertices with degree in a given set, thus extending results obtained by Aldous, Duquesne and Rizzolo.