# 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

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

Schedule for Fall 2008

Schedule for Winter 2008

Schedule for Winter 2007

Schedule for Fall 2006

Schedule for Winter 2006

Schedule for Fall 2005

Schedule for Winter 2005

Schedule for Fall 2004

Schedule for Winter 2004

Schedule for Fall 2003