Asymptotic properties of expansive Galton-Watson trees

Abstract

We consider a super-critical Galton-Watson tree $\tau $ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau $ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N} $. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. 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 $Z_n$. 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 $(\tau ^\theta , \theta \in [0, \infty ])$.