Annular noncrossing permutations and partitions, and second-order asymptotics for random matrices

Abstract

We study the set Sann−nc(p,q) of permutations of {1, …, p+q} which are noncrossing in an annulus with p points marked on its external circle and q points marked on its internal circle. We define Sann−nc(p,q,q) algebraically by identifying the crossing patterns which can occur in an annulus. We prove the annular counterpart for a “geodesic condition” shown by Biane to characterize noncrossing permutations in a disc. We examine the relation between Sann−nc(p,q,q) and the set NC ann (p,q of annular noncrossing partitions of {1, …, p+q} and observe that (unlike in the disc case) the natural map from Sann−nc(p,q) onto NC ann (p,q) has a pathology which prevents it from being injective. We point out that annular noncrossing permutations appear in the description of the second-order asymptotics for the joint moments of certain families (Wishart and GUE) of random matrices. Some of the formulas extend to a multiannular framework; as an application of that, we observe a phenomenon of asymptotic Gaussianity for traces of words made with independent Wishart matrices.