Open Access
November 2013 Limiting distribution of maximal crossing and nesting of Poissonized random matchings
Jinho Baik, Robert Jenkins
Ann. Probab. 41(6): 4359-4406 (November 2013). DOI: 10.1214/12-AOP781


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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Jinho Baik. Robert Jenkins. "Limiting distribution of maximal crossing and nesting of Poissonized random matchings." Ann. Probab. 41 (6) 4359 - 4406, November 2013.


Published: November 2013
First available in Project Euclid: 20 November 2013

zbMATH: 1303.60008
MathSciNet: MR3161478
Digital Object Identifier: 10.1214/12-AOP781

Primary: 60B10 , 60F99
Secondary: ‎33D45 , 35Q15

Keywords: crossing , nesting , orthogonal polynomials , Random matchings , random matrices , Riemann–Hilbert problems

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 6 • November 2013
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