Probabilistic Operator Algebra Seminar, Winter 2006

Schedule for Fall 2006 Thursday, June 15, 10:30 - 12:00 pm, Jeffery 422 Michael Neagu, Waterloo Asymptotic Freeness of Random Permutation Matrices with Restricted Cycle Lengths Let U_1, U_2, ... ,U_s be an independent family of uniformly distributed random N x N permutation matrices with cycle lengths restricted to given sets A_1, A_2, ... ,A_s of positive integers. Under fairly general conditions on the sets A_i, it is shown that this family converges in distribution as N -> infty to a *-free family (in the sense of Voiculescu) u_1, u_2, ... ,u_s, where each u_i is a Haar unitary of order sup A_i. Monday, April 3, 4:00 - 5:30 pm, Jeffery 319 Jonathan Novak, Queen's Title: The Harer-Zagier recursion formula and Gaussian matrix integrals ABSTRACT: For a positive integer n, consider a 2n-gon with edges labelled s_1,s_2,...,s_{2n}. Glueing the edges together in pairs, we obtain an orientable surface of genus g. For a given g, how many of the (2n)!! possible pairings will produce a surface of genus g? A recursion formula for this problem was derived by Harer and Zagier using random matrix techniques, and we will outline their argument.

Monday, March 27, 4:00 - 5:30 pm, Jeffery 319 Reza Rashidi Far, Queen's Title: Gaussian and Haar Unitary Random matrices Abstract: Applying combinatorics techniques in the free probability framework, we study the distribution of the eigenvalues of a random matrix. The Wick formula is used to calculate the moments of the eigenvalues of a selfadjoint Gaussian random matrix. Using this result, we show that the distribution of the eigenvalues when N goes to infinity, goes asymptotically to a semicircular (Wigner) distribution. Then we study the asymptotic free independence for several Gaussian random matrices as well as the asymptotic free independence between Gaussian random matrices and constant matrices. For Haar Unitary matrices, we do same by using the Weingarten function and utilizing the length function on permutations. This talk is based on Lecture 22 and 23 of the book, "Lectures on the Combinatorics of Free Probability theory." by A. Nica and R. Speicher.

Friday, March 17, 4:00 pm - 6:00 pm, Jeff 116 NOTE CHANGE OF TIME Claus Koestler, Carleton On coboundaries of non-commutative continuous Bernoulli shifts. Abstract: Recent work of Tsirelson shows that 'non-classical' stochastic flows provide a rich source of continuous product systems of Hilbert spaces of non-type I. These product systems have been introduced by Arveson to classify one-parameter semigroups of endomorphisms on B(H). Such product systems are of type I if and only if the corresponding semigroup is cocycle conjugated to the CAR/CCR shift. Independently, and based on operator algebraic techniques, some further non-type I examples are known from the work of Powers. But up to present, it is unknown whether such systems can be produced in a framework of non-commutative probability. We address this problem by the investigation of circular Brownian motion. Circular Brownian motion is a stationary T-valued process with unitary Brownian motions as increments. It provides a simple example for 'non-classical' stochastic flows, in the sense of Tsirelson. We identify circular Brownian motion as a coboundary with respect to the group action of point transformations on the underlying probability space and transfer its notion to the operator-algebraic setting of non-commutative continuous Bernoulli shifts. I prove, in particular, that a non-commutative white noise has only trivial coboundaries.

Monday, March 13, 4:00 - 5:30, Jeff 319 Wlodek Bryc, Cincinnati Title: Spectral measure of large random Hankel and Toeplitz matrices Abstract: This talk is based on the joint paper with with A. Dembo and T. Jiang in which we study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables of unit variance, scaling the eigenvalues by the square root of the dimension we prove the almost sure, weak convergence of the spectral measures to universal, non-random, symmetric distributions of unbounded support. Simulations suggest that both limiting laws have smooth densities and that the limiting law of Hankel matrices is bi-modal.

Monday, February 27, 4:00 - 5:30, Jeff 319 Benoit Collins, Ottawa/Lyons Title: Integration over compact quantum groups. Abstract: We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures. This is a joint work with Teodor Banica.

Monday, February 13, 4:00 - 5:30, Jeff 319 Helene Massam, York TITLE: Wishart distributions for graphical models ABSTRACT: The Wishart distribution is one of the fundamental distributions in classical multivariate statistics. In recent years graphical Gaussian models have proved very useful for the analysis of complex high-dimensional multivariate data. Distributions derived from the Wishart have been defined as the distribution of the maximum likelihood estimate or as prior distribution for parameters in these models. We will show how these new families of distributions can be extended and viewed as natural exponential families and how these extensions can be used themselves as more flexible prior distributions for graphical Gaussian models.

Monday, February 6, 4:00 -- 5:30 p.m. Jeffery 319 Emily Redelmeier, Queen's Explicit Construction of an Orthogonal Basis for the Temperley-Lieb Algebra Abstract: In this talk, we will discuss the orthogonal basis for the Temperley-Lieb algebra constructed by Genauer and Stoltzfus. An explicit recursion formula is given for the elements of the orthogonal basis which makes use of the Chebyshev polynomials. This can be used to demonstrate the positivity of the Markov form and the connection between the roots of the Chebyshev polynomial and the values of the quantum parameter q for which the Markov form is non-degenerate. This talk will concentrate on the chord diagram manipulations involved in the construction.

Monday, January 30, 4:00 - 5:30 pm, Jeff 319 Jamie Mingo, Queen's Fluctuations of Gaussian and Wishart Random Matrices Two well understood and widely used ensembles of random matrices are the Gaussian (also called GUE) and Wishart ensembles. The limiting values of their eigenvalue distributions are given respectively by the Wigner (semi-circle) law and the Marchenko-Pastur law. The fluctuations we have in mind are the covariances of eigenvalues. We shall work out the corresponding limiting distributions in each of the two cases mentioned above. We shall also give a general technique for finding the limiting distribution given the covariance of the resolvents. This can be viewed as Steiltjes inversion for a second order Cauchy transform. This is joint work with Roland Speicher.

Monday, January 23, 4:00 - 5:30 pm (note the shift in time!), Jeff 319 Roland Speicher (Queen's) TITLE: Free Probability Theory and Strong Haagerup Inequalities (joint work with T. Kemp) Abstract: In a holomorphic context, some analytic inequalities improve when restricted to a holomorphic subalgebra. We are exploiting a similar phenomena in a non-commutative situation. Concretely, we show that one can improve Haagerup's classical inequality for norms of convolution operators on the free group if one restricts to operators which only involve the generators (but not their inverses!) of the free group. Actually, the same type of statement is true in a much more general situation, namely for algebras generated by free R-diagonal elements. The proof of these statements relies on a good understanding of moments of the involved operators and uses the combinatorial machinery of free probability. At least the first half of the talk will give an introduction to the problem and the idea of the proof and does not suppose any knowledge about free probability theory. Monday, January 16, 4:30 - 6:00, Jeff 319 Serban Belinschi, Waterloo TITLE: Time behaviour for free convolution semigroups and free Brownian motion ABSTRACT: We will discuss the free convolution semigroups of Nica and Speicher and the free Brownian motion from an analytic point of view. We start by briefly reviewing properties of measures belonging to such semigroups and comparing them with properties of measures which are a free convolution with a semicircular distribution, as they were determined by Biane. Using some simple analytical tools, we prove continuity with respect to the time variable at the origin in the topology of a.e. convergence. If time permits, we will also discuss some possible applications to free entropy. Part of the results presented are joint work with Hari Bercovici, and part are derived from ongoing joint work with Alice Guionnet and Andrei Okounkov.

Schedule for Fall 2005