$k$ -规则自相似片段中最大和最小片段的急剧集中
Sharp concentration for the largest and smallest fragment in a $k$-regular self-similar fragmentation
摘要：
研究了$k$ -正则自相似碎裂过程的渐近性。对于$α> 0$和整数$k \geq 2$，这是马尔可夫过程$(I_t)_{t \geq 0}$，其中每个$I_t$是$[0,1)$的开子集的并集，并且独立地，大小为$u$的$I_t$的每个子区间以$u^α$的速率分解成大小相等的$k$块。令$k^{ - m_t}$和$k^{ - M_t}$分别为$I_t$中最大和最小片段的大小。通过将$(I_t)_{t \geq 0}$与分支随机游动联系起来，我们发现存在明确的确定性函数$g(t)$和$h(t)$，使得对于所有足够大的$t$，都有$|m_t - g(t)| \leq 1$和$|M_t - h(t)| \leq 1$.此外，对于每个$n$，我们研究大小为$k^{-n}$的片段存在的最终时间。特别是，通过将我们的分支随机游走与某个点过程联系起来，我们表明，经过适当的重新缩放，这些时间的定律收敛于Gumbel分布，即$n \to \infty$。
英文摘要：
We study the asymptotics of the $k$-regular self-similar fragmentation process. For $α> 0$ and an integer $k \geq 2$, this is the Markov process $(I_t)_{t \geq 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$, and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally sized pieces at rate $u^α$. Let $k^{ - m_t}$ and $k^{ - M_t}$ be the respective sizes of the largest and smallest fragments in $I_t$. By relating $(I_t)_{t \geq 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t - g(t)| \leq 1$ and $|M_t - h(t)| \leq 1$ for all sufficiently large $t$. Furthermore, for each $n$, we study the final time at which fragments of size $k^{-n}$ exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as $n \to \infty$.
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