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

By James A. Mingo, Alexandru Nica. 2003

97 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/923c295693576f9588d1eb1d2f28df3fb340f07d

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Citations in the corpus

GLOBAL FLUCTUATIONS FOR LINEAR STATISTICS OF β-JACOBI ENSEMBLES (2012, background): “Other approaches to calculating fluctuations for linear functionals for β-ensembles include the Capitaine and Casalis work [12], which, through free probability, obtains results for both Wishart and Jacobi (Double Wishart) matrices in the case β = 2.”
Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants (2006, background): “Global fluctuations of the eigenvalues (that is, linear functionals of the eigenvalues) of random matrices have been widely investigated in the last decade; see, e.g., [Joh98, Dia03, Rad04, MN04 , M ´ Roughly speaking, the trend of these investigations is that for a wide class of converging random ”
Second order freeness and fluctuations of random matrices: II. Unitary random matrices (2004, unknown): “ion is not one-to-one. Since we will not use the language of annular partitions in the present paper, this is of no relevance here. Annular non-crossing permutations and partitions were introduced in [MN]; there, many diﬀerent characterizations—in particular, the one (2) above in terms of the length”
Free Probability and Random Matrices (2014, methodology): “In [126] a permutation of [p+q] was defined to be a noncrossing annular permutation if no one of five proscribed configurations appeared.”
Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces (2004, methodology): “The proof of this theorem relies on the same kind of calculations as, for example, in [ MN ].”
Cleaning large correlation matrices: tools from random matrix theory (2016, background): “ (3.5) and is independent from C. The model falls into the model of free multiplication encountered in Section2.3since E is the free multiplicative convolution of C with a white Wishart kernel for N!1[108]. Therefore, the Stieltjes transform of E is exactly given by Eq. (2.84) that we specialize to ”