The Annals of Probability

Limiting distribution of maximal crossing and nesting of Poissonized random matchings

Jinho Baik and Robert Jenkins

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Abstract

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.

Article information

Source
Ann. Probab., Volume 41, Number 6 (2013), 4359-4406.

Dates
First available in Project Euclid: 20 November 2013

Permanent link to this document
https://projecteuclid.org/euclid.aop/1384957791

Digital Object Identifier
doi:10.1214/12-AOP781

Mathematical Reviews number (MathSciNet)
MR3161478

Zentralblatt MATH identifier
1303.60008

Subjects
Primary: 60B10: Convergence of probability measures 60F99: None of the above, but in this section
Secondary: 33D45: Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.) 35Q15: Riemann-Hilbert problems [See also 30E25, 31A25, 31B20]

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

Citation

Baik, Jinho; Jenkins, Robert. Limiting distribution of maximal crossing and nesting of Poissonized random matchings. Ann. Probab. 41 (2013), no. 6, 4359--4406. doi:10.1214/12-AOP781. https://projecteuclid.org/euclid.aop/1384957791


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