泊松化随机匹配的最大交叉极限分布与嵌套
Limiting distribution of maximal crossing and nesting of Poissonized random matchings
摘要：
Chen等人[Trans.Amer.Math.Soc.359（2007）1555-1575]引入了完全匹配的$r$ -交叉和$r$ -嵌套的概念，并证明了对称性。我们考虑随机匹配的大尺寸和研究他们的最大交叉和最大嵌套。已知它们中的每一个的边缘分布收敛于GOE Tracy-Widom分布。我们表明，最大交叉和最大嵌套成为独立的渐近，我们评估的联合分布的泊松随机匹配明确的第一个修正项。这导致了协方差的渐近评估。此外，我们计算的分布函数中的两个对象的显式第二个校正项：（a）的泊松随机置换的最长增加子序列的长度和（b）的最大交叉，因此也是最大嵌套，泊松随机匹配。
英文摘要：
The notion of $r$-crossing and $r$-nesting of a complete matching was introduced and a symmetry property was proved by Chen et al. [Trans. Amer. Math. Soc. 359 (2007) 1555-1575]. We consider random matchings of large size and study their maximal crossing and their maximal nesting. It is known that the marginal distribution of each of them converges to the GOE Tracy-Widom distribution. We show that the maximal crossing and the maximal nesting becomes independent asymptotically, and we evaluate the joint distribution for the Poissonized random matchings explicitly to the first correction term. This leads to an evaluation of the asymptotic of the covariance. Furthermore, we compute the explicit second correction term in the distribution function of two objects: (a) the length of the longest increasing subsequence of Poissonized random permutation and (b) the maximal crossing, and hence also the maximal nesting, of Poissonized random matching.
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