James A. Mingo

Second order freeness and fluctuations of random matrices: II. Unitary random matrices

Abstract We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of “second order freeness”, which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large N limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.

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

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.

Free Probability and Random Matrices

The concept of freeness was introduced by Voiculescu in the context of operator algebras. Later it was observed that it is also relevant for large random matrices. We will show how the combination of various free probability results with a linearization trick allows to address successfully the problem of determining the asymptotic eigenvalue distribution of general selfadjoint polynomials in independent random matrices.

ORTHOGONAL POLYNOMIALS AND FLUCTUATIONS OF RANDOM MATRICES

Abstract In this paper we establish a connection between the fluctuations of Wishart random matrices, shifted Chebyshev polynomials, and planar diagrams whose linear spans form a basis for the irreducible representations of the annular Temperly-Lieb algebra.

Real second order freeness and Haar orthogonal matrices

We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier.

Sharp bounds for sums associated to graphs of matrices

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form Sπ(N):=∑j1,…,j2m=1kerj⩾πNtj1j2(1)tj3j4(2)⋯tj2m−1j2m(m) where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph Gπ associated to the constraints, π, in the sum.

Operator Algebras and Their Applications

A generalized intertwining lifting theorem by B. V. Rajarama Bhat On the classification of C*-algebras of real rank zero, III: The infinite case by O. Bratteli, G. A. Elliott, D. E. Evans, and A. Kishimoto On the classification of C*-algebras of real rank zero, IV: Reduction to local spectrum of dimension two by G. A. Elliott, G. Gong, and H. Su Simple approximate circle algebras by I. Stevens The classification of certain non-simple approximate interval algebras by K. H. Stevens Right inverse of the module of approximately finite dimensional factors of type III and approximately finite ergodic principal measured groupoids by C. E. Sutherland and M. Takesaki.

Second order cumulants of products

We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of Krawczyk and Speicher. We apply our result to the problem of calculating the second order cumulants of a semi-circular and Haar unitary operator.

C*-Algebras associated with one dimensional almost periodic tilings

For each irrational number, 0<α<1, we consider the space of one dimensional almost periodic tilings obtained by the projection method using a line of slope α. On this space we put the relation generated by translation and the identification of the “singular pairs”. We represent this as a topological spaceXα with an equivalence relationRα. OnRα there is a natural locally Hausdorff topology from which we obtain a topological groupoid with a Haar system. We then construct the C*-algebra of this groupoid and show that it is the irrational rotation C*-algebra,Aα.

On the Distribution of the Area Enclosed by a Random Walk onZ2

Let?2nbe the set of paths with 2nsteps of unit length inZ2, which begin and end at (0,0). For???2n, letarea(?)?Zdenote the oriented area enclosed by?. We show that for every positive even integerk, there exists a rational functionRkwith integer coefficients, such that:1|?2n|???Gamma;2n[area(?)]k=Rk(n,n>2k.We calculate explicitly the degree and leading coefficient ofRk. We show how as a consequence of this (and by also using the enumeration of up-down permutations, and the exponential formula for cycles of permutations) one can derive the asymptotic distribution of the area enclosed by a random path in?2n. The formula for the asymptotic distribution can be stated as follows: for?