On the maximal displacement of subcritical branching random walks with or without killing

Speaker: 侯浩杰（北理工）

Title: On the maximal displacement of subcritical branching random walks with or without killing

Inviter: 随机分析研究中心

Time & Venue: 2025年9月12日16:00--17:00 南楼613

Abstract: Consider a subcritical branching random walk $\{Z_k\}_{k\geq 0}$ with offspring distribution $\{p_k\}_{k\geq 0}$ and step size $X$. Let $M_n$ denote the rightmost position reached by $\{Z_k\}_{k\geq 0}$ up to generation $n$ and define $M := \sup_{n\geq 0} M_n$. In this talk, we give asymptotics of tail probability of $M$ under optimal assumptions $\sum^{\infty}_{k=1}(k\log k) p_k<\infty$ and $\mathbb{E}[Xe^{\gamma X}]<\infty$, where $\gamma >0$ is a constant such that $\mathbb{E}[e^{\gamma X}]=\frac{1}{m}$ and $m=\sum_{k=0}^\infty kp_k\in (0,1)$. Moreover, we confirm the conjecture of Neuman and Zheng [Probab. Theory Related Fields, 2017] by establishing the existence of a critical value $m\mathbb{E}[X e^{\gamma X}]$ such that \begin{align} \lim_{n\to\infty} \mathbb{P}(M_n\geq cn\big| M\geq cn)= \begin{cases} &1,~c\in\big(0,m\mathbb{E}[Xe^{\gamma X}]\big); \\ &0,~c\in\big(m\mathbb{E}[Xe^{\gamma X}], \infty\big). \end{cases} \end{align} Finally, we extend these results to the maximal displacement of branching random walks with killing. Based on a joint work with Shuxiong Zhang (Anhui Normal University).