AI 中文总结
该研究将逆序图最小度数定义为最小逆序，推广了相关定理，证明其可渐近缩放为n^β（β∈[0,1]），且概率呈指数衰减规律。
AI 中文摘要
给定一个排列σ，其对应的逆序图定义为：当且仅当i<j且σ(i)>σ(j)时，在i和j之间添加一条边。关于随机逆序图的早期研究成果来自Acan和Pittel，他们研究了具有固定逆序数的均匀排列的连通阈值；Bhattacharya与Mukherjee则主要关注均匀随机排列对应的逆序图的度数。在本研究中，我们将逆序图的最小度数称为“最小逆序”，并将Bhattacharya与Mukherjee的定理推广至如下情形：排列不仅是均匀的，还可由平面上某分布采样得到的点的排序生成。在该分布满足正则性假设，且存在合适的α>0时，我们证明：最小逆序经n^(α/(α+1))缩放后大于t的概率，表现为exp(-ct^(α+1))，其中c>0是依赖于该分布的常数。我们进一步证明，对任意α>0，至少存在一个对应的分布，从而证明最小逆序可渐近缩放为n^β，其中β∈[0,1]（β=0和β=1的情形可通过恒等排列、逆恒等排列等实现）。
英文摘要
Given a permutation $σ$, its corresponding \textit{inversion graph} is obtained by adding an edge between $i<j$ if and only if $σ(i)>σ(j)$. The first results on random inversion graphs come from Acan and Pittel, who studied the connected threshold for a uniform permutation with fixed inversion number, and Bhattacharya and Mukherjee, who mostly focused on the degrees of the graph when the permutation is chosen uniformly at random. In this work, we call \textit{minimal inversion} the minimal degree of the inversion graph and extend a theorem from Bhattacharya and Mukherjee to the case where the permutation is not only uniform, but obtained as the ordering of points sampled according to some distribution on the plane. Under regularity assumptions on the distribution, and for the appropriate $α>0$, we show that the probability that the minimal inversion rescaled by $n^{α/(α+1)}$ is larger than $t$ behaves like $\exp(-ct^{α+1})$ for some constant $c>0$ depending on the distribution. We further show that every $α>0$ admits at least one corresponding distribution, thus proving that the minimal inversion can asymptotically scale as $n^β$ for any $β\in[0,1]$ (the cases $β=0$ and $β=1$ being obtained via the identity and anti-identity permutations, among others).
Comments21 pages, 3 figures
