Asymptotic properties of expansive Galton-Watson trees
北京师范大学随机数学中心学术报告
报告题目(Title)：Asymptotic properties of expansive Galton-Watson trees
报告人(Speaker)：Jean-Francois DELMAS (Ecole des Ponts ParisTech)
地点(Place)：后主楼1220B
时间(Time)：2019年11月19日 16:00-17:00
邀请人(Inviter)：何辉
报告摘要
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 $a_n\in {\mathcal N}$. 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 $(\tau^\theta, \theta\in [0, \infty])$.
(Joint work with Romain Abraham, published in Electron. J. Probab. 24 (2019))
