随机置换图的度序列
Degree Sequence of Random Permutation Graphs
摘要：
本文研究了随机置换序列$π_n\in S_n$所对应的置换图$G_{π_n}$的度序列。联合极限分布的程度建立使用图和置换极限理论的结果。特别地，对于均匀随机排列，标记为$\lceil nr_1 \rceil, \lceil nr_2 \rceil, \ldots, \lceil nr_s \rceil$的顶点的度的联合分布（在通过$n$缩放之后）收敛到独立随机变量$D_1, D_2, \ldots, D_s$，其中对于$r_i\in [0,1]$和$i\in \{1, 2, \ldots, s\}$，为$D_i\sim \text{Unif}(r_i, 1-r_i)$。此外，中间顶点（标记为$n/2$的顶点）的度具有中心极限定理，并且在适当的缩放后，最小度收敛于Rayleigh分布。最后，与Mallow随机置换相关的置换图的极限度分布被确定，并观察到有趣的相变。我们的结果扩展到其他指数排列的措施。
英文摘要：
In this paper we study the degree sequence of the permutation graph $G_{π_n}$ associated with a sequence $π_n\in S_n$ of random permutations. Joint limiting distributions of the degrees are established using results from graph and permutation limit theories. In particular, for the uniform random permutation, the joint distribution of the degrees of the vertices labelled $\lceil nr_1 \rceil, \lceil nr_2 \rceil, \ldots, \lceil nr_s \rceil$ converges (after scaling by $n$) to independent random variables $D_1, D_2, \ldots, D_s$, where $D_i\sim \text{Unif}(r_i, 1-r_i)$, for $r_i\in [0,1]$ and $i\in \{1, 2, \ldots, s\}$. Moreover, the degree of the mid-vertex (the vertex labelled $n/2$) has a central limit theorem, and the minimum degree converges to a Rayleigh distribution after appropriate scalings. Finally, the limiting degree distribution of the permutation graph associated with a Mallows random permutation is determined, and interesting phase transitions are observed. Our results extend to other exponential measures on permutations.
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