solidot新版网站常见问题,请点击这里查看。
消息
本文已被查看7765次
China’s facial-recognition startups can probably pick you out of a crowd
来源于:MIT Technology
In this paper, we consider the soliton cellular automaton introduced in
[Takahashi 1990] with a random initial configuration. We give multiple
constructions of a Young diagram describing various statistics of the system in
terms of familiar objects like birth-and-death chains and Galton-Watson
forests. Using these ideas, we establish limit theorems showing that if the
first $n$ boxes are occupied independently with probability $p\in(0,1)$, then
the number of solitons is of order $n$ for all $p$, and the length of the
longest soliton is of order $\log n$ for $p<1/2$, order $\sqrt{n}$ for $p=1/2$,
and order $n$ for $p>1/2$. Additionally, we uncover a condensation phenomenon
in the supercritical regime: For each fixed $j\geq 1$, the top $j$ soliton
lengths have the same order as the longest for $p\leq 1/2$, whereas all but the
longest have order at most $\log n$ for $p>1/2$. As an application, we obtain
scaling limits for the lengths of the $k^{\text{th}}$ longest increasing and 查看全文>>