2011/11/30 by Jinho Baik, Robert Jenkins · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Asymptotic distribution #Bayesian Methods and Mixture Models #Combinatorics #Discrete mathematics #Distribution (mathematics) #Joint probability distribution #Matching (statistics) #Mathematical analysis #Mathematics #Nesting (process) #Permutation (music) #Random Matrices and Applications #Random permutation #Statistics #math-ph #math.CO #math.MP #math.PR
paper · pdf · doi:10.1214/12-aop781
published as Annals of Probability 2013, Vol. 41, No. 6, 4359-4406 · Published in at http://dx.doi.org/10.1214/12-AOP781 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2013/11/01 · arxiv created 2013/12/16 · arxiv updated 2015/03/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.