Limiting distribution of maximal crossing and nesting of Poissonized random matchings

TL;DR

This paper investigates the asymptotic behavior of maximal crossing and nesting in large random matchings, showing their independence and providing explicit joint distribution corrections.

Contribution

It establishes the asymptotic independence of maximal crossing and nesting and computes explicit correction terms for their joint distribution in Poissonized matchings.

Findings

Maximal crossing and nesting become asymptotically independent.

Explicit first and second correction terms for joint distribution are derived.

Asymptotics of longest increasing subsequence length in Poissonized permutations are computed.

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.