Asymptotic properties of expansive Galton-Watson trees

TL;DR

This paper investigates the asymptotic local limits of super-critical Galton-Watson trees conditioned on large generation sizes, identifying different regimes and their corresponding limit trees, with new results in low and high regimes.

Contribution

It characterizes the local limits of Galton-Watson trees in various growth regimes, including new results for low and high regimes, and explores their continuity properties.

Findings

Identified local limits as Kesten trees in the low regime.

Described the family of local limits conditioned on the limit of normalized generation size.

Proved convergence towards a limit tree in the Harris case with finite offspring support.

Abstract

We consider a super-critical Galton-Watson tree whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau$n distributed as $\tau$ conditioned on the n-th generation, Zn, to be of size an $\in$ N. We identify the possible local limits of $\tau$n as n goes to infinity according to the growth rate of an. In the low regime, the local limit $\tau$ 0 is the Kesten tree, in the moderate regime the family of local limits, $\tau$ $\theta$ for $\theta$ $\in$ (0, +$\infty$), is distributed as $\tau$ conditionally on {W = $\theta$}, where W is the (non-trivial) limit of the renormalization of Zn. In the high regime, we prove the local convergence towards $\tau$ $\infty$ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits ($\tau$ $\theta$ , $\theta$ $\in$ [0, $\infty$]).