A complete matching of [2n] is a pairing {(i_1, j_1), ... , (i_n,
j_n)} of 2n elements and can be thought of as a permutation on 2n
consisting of only 2-cycles. Stanley et al. introduced the notion of
k-crossing and k-nesting matchings and showed that the marginal
distribution of each converges to the GOE Tracy-Widom distribution.
Together with Jinho Baik, we showed that the maximal crossing and
nesting are asymptotically independent and evaluate the the joint
distribution of Poissonized random matchings up to the first correction.
In this talk, we will introduce the above concepts, give an
interpretation of the joint distribution of Poissonized random
matchings in terms of non-intersecting walks, and show how this joint
distribution can be evaluated in terms of certain discrete orthogonal
polynomials on the unit circle. Time permitting, I'll mention some
other results including an interesting connection to the
Ablowitz-Ladik equation.