LIMITING DISTRIBUTION OF MAXIMAL CROSSING AND NESTING OF POISSONIZED RANDOM MATCHINGS

• 成果类型: Article
• 署名作者: Baik, Jinho; Jenkins, Robert
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP781
• 发表日期: 2013
• 页码: 4359-4406
• 关键词: longest increasing subsequence; tracy-widom limits; LARGEST EIGENVALUE; asymptotics; UNIVERSALITY; CONVERGENCE; POLYNOMIALS; BOUNDARY; Respect; GROWTH

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.