梳状图上碰撞数的相变研究
On the phase transition for the number of collisions on comb graphs
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- National University of Singapore（新加坡国立大学）
- The Hong Kong University（香港大学）
- The Hong Kong University of Science and Technology（香港科技大学）
- Technische Universität München（慕尼黑工业大学）
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中文总结 AI 辅助
该研究分析梳状图上简单随机游走碰撞数的相变，确定不同轮廓参数γ下两/三个独立随机游走发生无限次或有限次碰撞的临界条件。
中文摘要 AI 辅助
我们研究梳状图$\boldsymbol{\text{Comb}}(\boldsymbol{\text{Z}},H)$上简单随机游走的碰撞，这类图是通过在整数轴的任意点$x$上附加形如$[0,H_x] \bigcap \boldsymbol{\text{Z}}$的垂直线段得到的。对于轮廓为$H_x(x) = |x| \boldsymbol{\text{log}}^\boldsymbol{\text{γ}}(|x| \bigvee 1)$的$\boldsymbol{\text{Comb}}(\boldsymbol{\text{Z}},H)$，我们证明：若$\boldsymbol{\text{γ}} \boldsymbol{\boldsymbol{\text{≤}}} \boldsymbol{2}$，则从同一点出发的两个独立简单随机游走几乎必然发生无限次碰撞。若齿轮廓取为满足$\boldsymbol{P}(H_x > z) \boldsymbol{\text{~}} Cz^{-\boldsymbol{\text{γ}}}$（其中$C > 0$，当$z$趋于无穷时）的独立同分布重尾随机变量的典型实现，我们证明：当$\boldsymbol{\text{γ}} \boldsymbol{\text{>}} \boldsymbol{1/3}$时，两个独立随机游走几乎必然发生无限次碰撞；当$\boldsymbol{\text{γ}} \boldsymbol{\text{∈}} (0,1/3)$时，几乎必然发生有限次碰撞；且对于任意$\boldsymbol{\text{γ}} \boldsymbol{\text{∈}} (0,1]$，三个独立随机游走几乎必然仅发生有限次碰撞。
英文摘要
We consider collisions of simple random walks on comb graphs $\mathrm{Comb}(\mathbb{Z},H)$, which are obtained by attaching vertical segments of the form $[0,H_x] \cap \mathbb{Z}$ to any point $x$ of the integer axis. For $\mathrm{Comb}(\mathbb{Z},H)$ with profile $H_x(x) = |x| \log^γ(|x| \vee 1)$, we show that two independent simple random walks starting from the same site collide infinitely often almost surely if $γ\leq 2$. If the tooth profile is taken as a typical realization of i.i.d. heavy-tailed random variables with $\textbf{P}(H_x > z) \sim Cz^{-γ}$ (with some $C > 0$) as $z$ tends to infinity, we show that infinitely many collisions occur almost surely for two independent random walks if $γ> 1/3$, whereas finitely many collisions occur almost surely if $γ\in (0,1/3)$, and for any $γ\in (0,1]$, three independent random walks only collide finitely many times, almost surely.
