Seminar on Free Probability and Random Matrices
Winter 2009

Organizers: J. Mingo and R. Speicher

Schedule for Fall 2009

Tuesday, April 21, 4:30 - 6:00, Jeff 319

Emily Redelmeier, (Queen's)

A Quaternionic Wick Formula

The Wick formula allows us to compute the expected values of
products of Gaussian random variables using a combinatorial
formula.  We will derive a version of this formula for
quaternionic Gaussian random variables, adapting the
combinatorial machinery for calculating moments of random
matrices to handle the noncommutativity of the quaternions.

Tuesday, April 14, 4:30 - 6:00, Jeff 422 (Room Change) Teodor Banica, (Toulouse) Free quantum groups - an overview For certain compact groups of matrices G_M_n(C), one can construct by the means of functional analysis and noncrossing partitions a quantum group G^+: the free version of G. The study of G^+, and of the correspondence G\to G^+, can be done via an interesting mix of algebraic and analytic methods. I will review the story of the subject, starting from its discovery by Wang 15 years ago, and ending with some recent results.

Tuesday, April 7, 4:30 - 6:00, Jeff 319 Matthew Lewis, (Queen's) Two Point Correlation Functions for the Gaussian Unitary Ensemble I will discuss the limiting form for R_1(x) R_1(y) - R_2(x,y) found by Cabanal-Duvillard and others.

Tuesday, March 31, 4:30 - 6:00, Jeff 319 Mike Brannan, (Queen's) Strong Haagerup Inequalities with Operator Coefficients Let k be a natural number, and let F_k denote the free group on k generators. The (classical) Haagerup inequality states that if g is a complex function supported on the set of words in F_k of length d, then the norm of g as a convolution operator on the Hilbert space l^2(F_k) is always dominated by (d+1)||g||_2, where ||g||_2 is the norm of g in l^2(F_k). In this talk I will briefly introduce the Haagerup inequality as well as a remarkable generalization and strengthening of this inequality (due to Todd Kemp and Roland Speicher) to the setting of non self-adjoint algebras generated by free R-diagonal elements. I will then go over a recent preprint of Mikael de la Salle, which generalizes the Kemp-Speicher strong Haagerup inequality to the operator space setting.

Wednesday, March 25, 1:00 - 2:30, Jeff 202 Richard Burstein (Ottawa) Automorphisms of the bipartite graph planar algebra A planar algebra is a graded vector space along with a certain graphical calculus, namely an associative action of the planar operad. These algebras were first developed by Jones for use in the classification of II_1 subfactors, but they have since been used in other areas such as category theory. The standard invariant of every (finite-index, extremal) II_1 subfactor may be described as a planar algebra; conversely, every planar algebra obeying certain additional conditions ("of subfactor type") is in fact the standard invariant of a subfactor. Planar algebras may also be obtained from bipartite graphs. These bipartite graph planar algebras are never of subfactor type, but they may have subfactor-type planar subalgebras. In fact every subfactor planar algebra may be embedded in the bipartite graph planar algebra on the subfactor's principal graph. In general, it is difficult to show that a graded subspace of a bipartite graph planar algebra P is closed under the planar operad. However, if we consider a group G of automorphisms of P (i.e., invertible graded linear maps on P which commute with the planar operad), then the set of fixed points P^G is closed under the operad action. I will describe the automorphism group of an arbitary bipartite graph planar algebra, and give conditions for the fixed points P^G to be of subfactor type. I will describe several examples of this construction, including some new infinite-depth subfactors.

Tuesday, March 24, 4:30 - 6:00, Jeff 319 Jiun-Chau Wang (Queen's) On the convergence to the semicircle law We first briefly review the history of the free central limit theorem. Then we will present a local estimate for the densities and an entropic central limit theorem. The source for these new results is complex analysis.

Tuesday, March 17, 4:30 - 6:00, Jeff 319 Jonathan Novak (Queen's) Contraction matrix models and the strong Szego-Bump-Diaconis limit theorem On can construct random contraction matrices by deleting rows and columns from Haar distributed random unitary matrices. Perhaps surprisingly, the resulting deformed model retains an exactly solvable structure. A natural question is then what other nice features of Haar unitaries are preserved by this deformation. It can be shown that traces of these random contractions are closely related to a statistical mechanical model of interacting particles on a one-dimensional lattice. Different scaling limits are then of interest: long term behaviour of the particle model is related to asymptotic moments of contractions of a fixed dimension, while letting the number of particles go to infinity corresponds to large matrix dimension for random contractions. The latter asymptotic can be got at using a strengthening of the classical Szego limit theorem for Toeplitz matrices due to Bump and Diaconis.

Tuesday, March 10, 4:30 - 6:00, Jeff 319 Jamie Mingo (Queen's) Fluctuations of Kesten's Law In 1959 Harry Kesten found the distribution of random walks on the free group on n generators, now known as Kesten's law. This law is the additive free convolution of the arcsine law with itself, n times. Kesten's law is also the limiting eigenvalue distribution of X_N = U_1 + U_1^{-1} +... + U_n + U_n^{-1} where {U_1 , ... , U_n } are independent N x N Haar distributed random unitary matrices. I shall present the limiting fuctuations of the random variables { Tr(X^k_N ) }_k and the orthogonal polynomials that diagonalize them.

Tuesday, March 3, 4:30 - 6:00, Jeff 319 Jack Silverstein, North Carolina State University Eigenvalues of Large Dimensional Random Matrices Abstract"

Tuesday, February 24, 4:30 - 6:00, Jeff 319 Roland Speicher (Queen's) Asmptotic freeness for Wigner matrices and estimates for graph sums Abstract: Whereas asymptotic freeness between Gaussian random matrices and deterministic matrices is by now a standard result, the generalization to Wigner matrices is, though generally believed to be true, not worked out in the literature. In this talk I will present a combinatorial approach to the asymptotic freeness between Wigner matrices and deterministic matrices; a special role in this approach will be played by estimates for sums which are defined in terms of graphs.

Tuesday, February 10, 4:30 - 6:00, Jeff 319 Roland Speicher (Queen's) The quantum permutation group I will give a light introduction into the quantum permutation group: its structure as quantum group, its Haar measure, and its (co-)actions. No prior knowledge on quantum groups is required.

Tuesday, February 3, 4:30 - 6:00, Jeff 319 Jamie Mingo (Queen's) Two and Three Point Correlation Functions for Eigenvalues of Random Matrices The k-point correlation functions of N x N random matrix describe the correlation of the eigenvalues of the random matrix. Even for the better understood ensembles, these functions are somewhat complicated. However in some cases simple expressions appear in the large N limit. In this talk (which is joint work with Roland Speicher) I will consider the case of the two and three point correlation functions of the Gaussian Unitary Ensemble. I will show that these have in the large N limit, a simple description in terms of planar diagrams. Explicit analytical formulas can then be derived using some divergent series.

Tuesday, January 27, 4:30 - 6:00, Jeff 319 Ricky Tang (Queen's) Random matrices: Universality of ESDs and the circular law Abstract: Given an n by n complex matrix A, we let mu_A(x,y) := 1/n |{1 < i < n, Re(lambda_i) < x, Im(lambda_i) < y}| be the empirical spectral distribution (ESD) of its eigenvalues lambda_i in C, i = 1, ... n. Then we have the well known Circular Law, which is stated as follows. Let X_n be the n by n random matrix whose entries are iid complex random variables with mean 0 and variance 1. It was known that with the extra assumption that the entries have finite (2 + epsilon)-th moment for any fixed epsilon > 0 the ESD of 1/sqrt(n) X_n converges to the uniform distribution on the unit disk. Recently Krishnapur, Tao, and Vu removed the (2 + epsilon) assumption. In my talk on the paper of Krishnapur, Tao, and Vu, I will present the so called The Replacement Principle, which gives a general criterion for the ESDs of the difference of two normalised random matrices 1/sqrt{n} A_n, 1/sqrt{n} B_n to converge to 0.

Tuedsday, January 20, 4:30 - 6:00, Jeff 319 Mihai Popa (Indiana University) On the conditionally free analogue of the $S$-transform Abstract: Using the combinatorics of non-crossing partitions, we construct a conditionally free analogue of the Voiculescu's S-transform. The construction is connected to non-crossing linked partitions and applied to analytical description of conditionally free multiplicative convolution and characterization of infinite divisibility.

Tuesday, January 13, 4:30 - 6:00, Jeff 319 Marco Bertola (Concordia) Cubic Strings, biorthogonal polynomials and Random Matrices Abstract. The spectral theory of a (classical) cubic string on a closed interval underlies the DeGasperis-Procesi hierarchy (akin to Camassa-Holms) and peakons. Although it is a non self-adjoint problem, the spectrum is real and positive but depends on boundary conditions. Projectors are in natural biorthogonality and are polynomials in the eigenvalue-parameter. These biorthogonal polynomials (suitably extended) depend on arbitrary positive measures supported on the positive real line. On a different approach, they arise from the study of a new two-matrix model over the product of the cones of positive matrices and the corresponding partition function (in the large-size limit) enumerates certain bicolored ribbon graph similar to the Itzykson-Zuber 2MM (but with different rules). Amongst other desirable features of the biorthogonal polynomials that we can establish at the moment are: 1) simplicity and positivity of the zeroes; 2) interlacing (total positivity of bimoment matrices); 3) Riemann-Hilbert characterization; 4) Riemann-Hilbert steepest-descent analysis for large degrees; 5) embedding as a reduction of the 2 Toda hierarchy.

Tuesday, January 6, 4:30 - 6:00, Jeff 319 Jonathan Novak (Queen's) The asymptotics of rectangular Young tableaux ABSTRACT: In a celebrated 1981 paper, Regev gave an asymptotic formula for the number of standard Young tableaux on the same shape contained in a strip of bounded height. By the Schensted correspondence, this is exactly the asymptotic number of permutations with bounded decreasing subsequence length. His approach relies on Selberg's integral formula. We will outline an elementary approach to Regev's result which relies on an asymptotic correspondence with rectangular Young tableaux. This approach, which is direct and avoids the use of Selberg's integral, has several advantages.

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