AI 中文总结
本文针对柱形Hastings--Levitov(0)聚集过程，证明其树完成时间期望与log N比值的极限为1/(2λ)，验证了前人提出的极限猜想。
AI 中文摘要
设CHL_N是宽度为N的柱面上、参数为0、粒子固定尺寸λ>0的柱形Hastings--Levitov聚集过程，ω_{N,λ}为其树完成时间，即基圆上新生树的最后时刻。Chen、Procaccia和Zong已证明上界E[ω_{N,λ}]≤(1+ε)(log N)/(2λ)并猜测匹配极限；本文证明匹配下界，故对所有固定λ>0，lim_{N→∞}E[ω_{N,λ}]/log N=1/(2λ)。
英文摘要
Let $\mathrm{CHL}_N$ be the cylindrical Hastings--Levitov aggregation process with parameter $0$ on a cylinder of width $N$ with particles of fixed size $λ>0$, and let $ω_{N,λ}$ be its tree-completion time --- the last time at which a new tree is born on the base circle. Chen, Procaccia and Zong proved the sharp upper bound $\mathbb{E}[ω_{N,λ}]\le(1+\varepsilon)(\log N)/(2λ)$ and conjectured the matching limit. Here we prove the matching lower bound, and therefore \[ \lim_{N\to\infty}\frac{\mathbb{E}[ω_{N,λ}]}{\log N}=\frac{1}{2λ} \qquad\text{for every fixed }λ>0 . \]
Comments10 pages, 1 figure
