  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.
