1111.0269 (Jinho Baik et al.)

Limiting distribution of maximal crossing and nesting of Poissonized
random matchings    [PDF]

Jinho Baik, Robert Jenkins

The notion of r-crossing and r-nesting of a complete match- ing was introduced and a symmetry property was proved by Chen, Deng, Du, Stanley and Yan in 2007. We consider random matchings of large size and study the maximal crossing and the 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 eval- uate the joint distribution for the Poissonized random matchings explic- itly to the first correction term. This leads to an evaluation of the asymptotic of the covariance. Furthermore, we compute the explicit second correction term of the distributions function of two ob jects: (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.

View original: http://arxiv.org/abs/1111.0269