
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES
July
2013
Focus Program on Noncommutative Distributions in Free
Probability Theory
July
26, 2013
Workshop on Combinatorial and Random Matrix Aspects
of Noncommutative Distributions and Free Probability
Organizing
Committee:
Serban Belinschi (Queen's), Alice Guionnet (MIT), Alexandru
Nica (Waterloo), Roland Speicher (Saarland)




8:45  9:15

Onsite Registration 
9:15  9:30

Welcome and Introduction 
9:30  10:30

DanVirgil Voiculescu, UC Berkeley (slides)
Free probability with left and right
variables 
10:30  11:00

Tea Break 
11:00  12:00

Greg Anderson, University of
Minnesota
Asymptotic freeness with little randomness
and the Weil representation of SL_2(F_p) 
12:00  2:00

Lunch Break 
2:00  3:00

Dimitri Shlyakhtenko, University
of California, Los Angeles
No atoms in spectral measures of
polynomials of free semicircular variables 
3:00  3:30

Tea Break 
3:30  4:00

Camille Male, Université
ParisDiderot (slides)
The spectrum of permutation invariant matrices 
4:00  4:30

Maxime Fevrier, Université
ParisSud 11
Outliers in the Spectrum of Spiked Deformations
of Unitarily Invariant Random Matrices 
4:30  5:00

Brendan Farrell, California
Institute of Technology (slides)
Structured Random Unitary Matrices and Asymptotic
Freeness 
Wednesday, July 3

9:30  10:30

Alan Edelman, Massachusetts
Institute of Technology
Isotropic Entanglement: A Fourth Moment
Interpolation Between Free and Classical Probability 
10:30  11:00

Tea Break 
11:00  12:00

Claus Koestler, University
College Cork (slides)
Quantum symmetric states on free product
C*algebras 
12:00  2:00

Lunch Break 
2:00  3:00

Philippe Biane, Universite ParisEst
(slides)
From noncrossing partitions to ASM 
3:00  3:30

Tea Break 
3:30  4:00

Mitja Mastnak, Saint Mary's
University (slides)
Twisted parking symmetric functions and free multiplicative
convolution 
4:00  4:30

JiunChau Wang, University
of Saskatchewan
Conservative Markov operators from 1D free harmonic analysis 
4:30  5:00

Roland Speicher, Saarland University (slides)
Selfadjoint polynomials in asymptotically
free random matrices 
Thursday, July 4

9:30  10:30

Alice Guionnet, Massachusetts
Institute of Technology
Heavy tails random matrices 
10:30  11:00

Tea Break 
11:00  12:00

Romuald Lenczewski, Wroclaw
University of Technology (slides)
Matricial freeness and random matrices 
12:00  12:30

Carlos Vargas, Universität
des Saarlandes (slides)
Block modifications of the Wishart ensemble and operatorvalued
free multiplicative convolution 
12:30  1:00

Ramis Movassagh, Northeastern
University (slides)
Isotropic Entanglement 
Friday, July 5

9:30  10:30

Alexandru Nica, University of
Waterloo
Some remarks on the combinatorics of free unitary Brownian motion 
10:30  11:00

Tea Break 
11:00  12:00

Jonathan Novak, Massachusetts
Institute of Technology
Asymptotics of Looped Cumulant Lattices 
12:00  2:00

Lunch Break 
2:00  3:00

Serban Belinschi, Queen's University
Spectral and Brown measures of polynomials in free random variables 
3:00  3:30

Tea Break 
3:30  4:00

Naofumi Muraki, Iwate Prefectural
University
On a qdeformation of free independence 
4:00  4:30

Emily Redelmeier, Université ParisSud
XI
Quaternionic SecondOrder Freeness 
4:30  5:00

Madhushree Basu, Institute of
Mathematical Sciences
Continuous CourantFischerWeyl minimax theorem 
Saturday, July 6

9:30  10:30

Mireille Capitaine, Université
Paul Sabatier (slides)
Exact separation phenomenon for the eigenvalues of large InformationPlusNoise
type matrices. Application to spiked models 
10:30  11:00

Tea Break 
11:00  12:00

Maciej Nowak, Jagiellonian University,
Cracow
Spectral shock waves in dynamical random matrix theories 
12:00  13:00

James Mingo, Queen's University
Asymptotic Freeness of Orthogonally and
Unitarily Invariant Ensembles 
Speaker

Title
and Abstract 
Anderson,
Greg
University of Minnesota 
Asymptotic freeness with little randomness and the Weil representation
of SL_2(F_p)
We try to make the case that the Weil (a.k.a. oscillator)
representation of SL_2(F_p) could be a good source of interesting (notvery)random
matrix problems.We do so by proving some asymptotic freeness results
and suggesting problems for research.
In particular, we offer an answer (by no means definitive) to a question
posed in a recent paper of Anderson and Farrell. We assume no familiarity
on the part of the audience with the Weil representation and will
explain how to construct it in downtoearth and explicit fashion.

Basu,
Madhushree
Institute of Mathematical Sciences 
Continuous CourantFischerWeyl minimax theorem
A minimax theorem is a result that gives a characterization
of eigenvalues of compact selfadjoint operators on Hilbert spaces.
The CourantFischerWeyl minmax theorem gives an extremum property of
the ${k^{th}}$ eigenvalue of a Hermitian $n \times n$ scalar matrix
(${1 \le k \le n}$), without referring to any eigenvector. Our work
grew out of the search for an extension of this theorem to a finite
von Neumann algebraic setting. We prove a version of this theorem for
a selfadjoint element having a non atomic distribution in a ${II_1}$
factor, and we also indicate an alternate proof for the finite dimensional
version of the original theorem. This noncommutative analogue of the
CFW minmax theorem uses the distribution function of the selfadjoint
operator as its main tool and makes use of a continuous version of Ky
Fan's theorem, which we state but do not prove in this talk. We finally
briefly discuss an application of the CFW theorem.
This is a joint work with V. S. Sunder.

Belinschi,
Serban
Queen's University

Spectral and Brown measures of polynomials in free random variables
The combination of a selfadjoint linearization trick due to
Greg Anderson with Voiculescu's subordination for operatorvalued free
convolutions and analytic mapping theory turns out to provide a method
for finding the distribution of any selfadjoint polynomial in free variables.
In this talk we will present the analytic machinery behind this process,
and show an extension that allows in principle the computation of Brown
measures of possibly nonselfadjoint polynomials in free variables.
We will also indicate some results on Brown measures of some sums and
products of free random variables. This talk is based on joint work
with Tobias Mai and Roland Speicher and ongoing joint work with Piotr
Sniady and Roland Speicher.

Biane,
Philippe
Universite ParisEst

From noncrossing partitions to ASM
Noncrossing partitions are related to alternating sign matrices
through the RazumovStroganov (ex)conjecture. I will review this as
well as attempts to relate ASMs with some plane partitions.

Capitaine,
Mireille
Université Paul Sabatier 
Exact
separation phenomenon for the eigenvalues of large InformationPlusNoise
type matrices. Application to spiked models
We consider large InformationPlusNoise type matrices of
the form ${M_N =(\sigma \frac{X_N}{\sqrt{N}}+A_N)(\sigma \frac {X_N}{\sqrt{N}}+A_N)^*}$
where ${X_N}$ is an ${n \times N (n \leq N)}$ matrix consisting of independent
standardized complex entries, ${A_N}$ is an ${n \times N}$ nonrandom
matrix and ${\sigma > 0}$. As N tends to infinity, if ${c_N = n/N
\rightarrow c \in [0, 1]}$ and if the empirical spectral measure ${\mu_{A_N{A_N}^*}}$
of ${A_N{A_N}^*}$ converges weakly to some compactly supported probability
distribution ${\nu \neq \delta_0}$, Dozier and Silverstein established
that almost surely the empirical spectral measure of ${M_N}$ converges
weakly towards a nonrandom distribution ${\mu_{\sigma,\nu,c}}$. Bai
and Silverstein proved, under certain assumptions on the model, that
for some fixed closed interval in ${]0;+\infty[}$ outside the support
of ${\mu_{\sigma,\mu_{A_N{A_N}^*},c_N}}$ for all large N, almost surely,
no eigenvalues of ${M_N}$ will appear in this interval for all N large.
We show that there is an exact separation phenomenon between the spectrum
of ${M_N}$ and the spectrum of ${A_N{A_N}^*}$: to a gap in the spectrum
of ${M_N}$ pointed out by Bai and Silverstein, it corresponds a gap
in the spectrum of ${A_N{A_N}^*}$ which splits the spectrum of ${A_N{A_N}^*}$
exactly as that of ${M_N}$. We deduce a relationship between the distribution
functions of some probability measures on ${\mathbb{R^+}}$ and their
rectangular free convolution with ratio c with the pushfoward of a MarchenkoPastur
distribution with parameter c by x${\mapsto \sqrt{x}}$. We use the previous
results to characterize the outliers of spiked InformationPlusNoise
type models.

Alan Edelman
Massachusetts Institute of Technology 
Isotropic Entanglement: A Fourth Moment Interpolation Between Free
and Classical Probability
The difference between the noncommutative and the commutative
moments of ABAB factor. This nifty little fact extends to the finite
dimensional case of random matrix theory allowing for a fourth moment
interpolation between free and classical probability that is suitable
for applications. We describe an application to a problem in quantum
many body physics, and mention comparisons with other interpolations
between free and classical probability. The method of ghosts and shadows
will be used and briefly discussed.
This is joint work with Ramis Movassagh.

Farrell,
Brendan
California Institute of Technology 
Structured Random Unitary Matrices and Asymptotic Freeness
A fundamental theorem of Voiculescu relating free probability
and random matrix theory states that conjugating deterministic matrices
by Haardistributed unitary matrices yields asymptotic freeness. In
work with Greg Anderson we show the existence of random unitary matrices
having more structure and less randomness yet also yielding asymptotic
freeness. We discuss how this work relates to discrete uncertainty principles
and classical random matrix theory.

Fevrier,
Maxime
Université ParisSud 11 
Outliers in the Spectrum of Spiked Deformations of Unitarily Invariant
Random Matrices
We investigate the asymptotic behavior of the eigenvalues
of the random matrix A+U*BU, where A and B are deterministic Hermitian
matrices and U is drawn from the unitary group according to Haar measure.
We discuss the existence and localization of "outliers", i.e.
eigenvalues lying outside from the bulk of the spectrum. This is joint
work with S. Belinschi, H. Bercovici and M. Capitaine.

Guionnet,
Alice
Massachusetts Institute of Technology 
Heavy tails random matrices
We will discuss properties of the sepctrum and eigenvectors
of random matrices with possibly large entries, such as the covariance
matrix of Erdos Renyi graphs or random matrices with alphastable entries.
This talk is based on joint works with Bordenave, BenaychGeorges and
Male.

Koestler,
Claus
University College Cork

Quantum symmetric states on free product C*algebras
Recently Roland Speicher and I had found a characterization
of freeness with amalgamation by quantum exchangeable random variables
in a W*algebraic setting of probability spaces. In this talk we introduce
quantum symmetric states in a C*algebraic setting of probability spaces
which extends the notion of quantum exchangeable random variables. Our
main result is a de Finetti type theorem for quantum symmetric states
and a characterization of extreme quantum symmetric states. We will
give some examples and, in particular, show that central quantum symmetric
states form a Choquet simplex whose extreme points are free product
states. Roughly speaking our results provide the free probability counterpart
of Stoermer's work on symmetric states on the infinite minimal tensor
product of a unital C*algebra. This is joint work with Ken Dykema and
John Williams.

Lenczewski,
Romuald
Wroclaw University of Technology

Matricial freeness and random matrices
I will discuss the concept of matricial freeness and its
applications to the study of limit distributions of independent random
matrices. In particular, I will show how to construct a random matrix
model for free Meixner laws and the associated ensemble.

Male,
Camille
Université ParisDiderot 
The spectrum of permutation invariant matrices

Mastnak,
Mitja
Saint Mary's University 
Twisted parking symmetric functions and free multiplicative convolution
The talk is based on joint work with A. Nica. I will describe
a correspondence between free multiplicative convolution of distributions
on a noncommutative probability space and convolution of characters
in the Hopf algebra of symmetric functions. The correspondence is established
via twisted parking symmetric functions. I will explain how these symmetric
functions can also arise from representation theory and mention some
connections with diagonal harmonics.

Mingo,
James Queen's University at Kingston 
Asymptotic Freeness of Orthogonally and Unitarily Invariant Ensembles
There has been a strong relation between unitary invariance
and asymptotic freeness ever since Voiculescu's 1991 paper on asymptotic
freeness. At the first order level there is very little difference between
the case of orthogonally and unitarily invariant ensembles. Above this
level the transpose plays a significant role in the orthogonal case,
something which isn't seen in the unitary case. This means one has to
consider ensembles ${\{A_N\}}$ in which there is joint limit distribution
for words in ${A_N}$ and ${A_N^t}$, i.e. a limit ${t}$distribution.
When one has an ensemble which has a limit ${t}$distribution and is
also unitarily invariant one gets the surprising result that ${A_N}$
and ${A_N^t}$ become free. In particular this applies to ensembles of
Haar distributed random unitary operators.
This is joint work with Mihai Popa.

Ramis
Movassagh
Northeastern University 
Isotropic Entanglement
The method of "Isotropic Entanglement" (IE), inspired
by Free Probability Theory and Random Matrix Theory, predicts the eigenvalue
distribution of quantum manybody (spin) systems with generic interactions.
At the heart is a "Slider", which interpolates between two
extrema by matching fourth moments. The first extreme treats the noncommuting
terms classically and the second treats them isotropically. Isotropic
means that the eigenvectors are in generic positions. We prove Matching
Three Moments and Slider Theorems and further prove that the interpolation
is universal, i.e., independent of the choice of local terms. Our examples
show that IE provides an accurate picture well beyond what one expects
from the first four moments alone.

Muraki,
Naofumi
Iwate Prefectural University 
On a qdeformation of free independence
I will introduce a product operation (qproduct) for noncommutative
probability spaces. There exists naturally a notion of `qindependence'
which is associated to the qproduct. Here an independence is understood
as a universal calculation rule for mixed moments of noncommutative
random variables, in the sense of Speicher but this time we drop the
associativity condition for such rule. The existence of universal calculation
rule for qproduct can be proved based on the card arrangement. The
qdeformed cumulants is also discussed.

Nica, Alexandru
University of Waterloo 
Some remarks on the combinatorics of free unitary Brownian motion
It is wellknown that, if ${u}$ is a Haar unitary, then the
joint free cumulants of ${u}$ and ${u^{*}}$ have a very nice explicit
description. The situation is not at all the same when instead of ${u}$
we look at a unitary ${u_t}$ (with ${t}$ in ${[0, \infty )}$) from the
free unitary Brownian motion. In my talk I will present some remarks
that can nevertheless be made about the joint free cumulants of ${u_t}$
and ${u_t^{*}}$. This is an ongoing joint work with Nizar Demni and
Mathieu GuayPaquet.

Novak,
Jonathan
Massachusetts Institute of Technology 
Asymptotics of Looped Cumulant Lattices
Since a noncommutative probability space is an algebra together
with one of its characters, the character description problem (given
${A}$, describe ${Char(A)}$) and character evaluation problem (given
${a \in A}$ and ${\tau \in Char(A)}$, compute ${\tau(a)}$) figure largely
in the theory. While explicit solutions of these problems are classically
known for many finitedimensional or almost finitedimensional algebras,
free probability asks us to solve these problems for free algebras.
I will describe the formalism of "looped cumulant lattices",
which produces special characters of free algebras that are computable
in terms of noncommutative differential operators. The name comes from
the fact that LCLs are often realizable as cumulants of polynomials
in random matrices.
This is based on ongoing work with Alice Guionnet.

Nowak,
Maciej
Jagiellonian University, Cracow 
Spectral shock waves in dynamical random matrix theories
We obtain several classes of nonlinear partial differential
equations for various random matrix ensembles undergoing Brownian type
of random walk. These equations for spectral flow of eigenvalues as
a function of dynamical parameter ("time") are exact for any
finite size N of the random matrix ensemble and resemble viscid Burgerslike
equations known in the theory of turbulence. In the limit of infinite
size of the matrix, these equations reduce to complex inviscid Burgers
equations, proposed originally by Voiculescu in the context of free
processes. We identify spectral shock waves for these equations in the
limit of the infinite size of the matrix, and then we solve exact, finite
N nonlinear equations in the vicinity of the shocks, obtaining in this
way universal, microscopic scalings equivalent to Airy, Bessel and corresponding
cuspoids (Pearcey, Bessoid) kernels.

Emily
Redelmeier Université ParisSud XI 
Quaternionic
SecondOrder Freeness
Secondorder freeness was created to extend free probabilistic approaches
to random matrices from their moments
to their global fluctuations. However, the natural definition for complex
random matrices, for which any independent,
unitarily invariant distributions are asymptotically secondorder free,
is no longer natural for real random matrices, which satisfy a different
definition, reflecting differences in the graphical calculuses for these
matrices. We discuss the natural definition for quaternionic random matrices.

Shlyakhtenko,
Dimitri
University of California, Los Angeles 
No atoms
in spectral measures of polynomials of free semicircular variables
In a joint work with P. Skoufranis, we show that if ${(X_j:1
\leq j \leq n)}$ are free variables with nonatomic spectral measures,
then ${Y=p(X_1,\dots,X_n)}$ has a nonatomic spectral measure, for any
nonconstant (matricial) polynomial ${p}$ in ${n}$ variables. The result
involves an extension of the Atiyah conjecture to products of certain
algebras.

Roland
Speicher
Saarland University 
Selfadjoint polynomials in asymptotically free random matrices
I will describe our recent progress on determining the asymptotic
eigenvalue distribution of polynomials in asymptotically free random
matrices. In the language of free probability this is the same as determining
the distribution of a polynomial in free variables. This problem was
solved in recent joint work with Tobias Mai and Serban Belinschi, by
further developing the analytic theory of operatorvalued additive free
convolution and combining this with Anderson's version of the linearization
trick.I will only address the case of selfadjoint polynomials. More
details and also extensions to the nonselfadjoint case will be presented
in the talk of Belinschi.

Carlos
Vargas
Universität des Saarlandes 
Block modifications of the Wishart ensemble and operatorvalued
free multiplicative convolution
We describe block modifications of $nd \times nd$ Wishart
matrices: For a selfadjoint linear map $\varphi : M_n (\mathbb{C})
\to M_n
(\mathbb{C})$, we consider the random matrix $W^{\varphi} := (id_d \otimes
\varphi)(W)$. We are interested in the asymptotic eigenvalue distribution
$\mu^{\varphi}$ as $d \to \infty$. The distributions of Wishart ensembles
were computed by Marchenko and Pastur. Voiculescu’s Free Probability
theory led to a new conceptual look of such laws, which can be seen
as the free Poisson distributions. The partial transpose ($\varphi(A)
= A^t$) was studied by Aubrun.
Later, Banica and Nechita recognized $\mu^{\varphi}$ as free compound
Poisson laws for a larger class of maps. In this talk we present an
operatorvalued free probabilistic approach. It turns out that \mu^{\varphi}
is exactly the matrixvalued free multiplicative convolution of a deterministic
matrix and a random diagonal matrix. We use the analytic subordination
approach to operator valued free multiplicative convolution to obtain
\mu^{\varphi} numerically, for any self adjoint map {\varphi}. This
talk is based in two joint works: with Belinschi, Speicher and Treilhard,
and with Arizmendi and Nechita.

Voiculescu,
DanVirgil
UC Berkeley

Free probability with left and right variables
I will describe the extension of free probability to systems
with left and right variables, based on a notion of bifreeness.

Wang,
JiunChau
University of Saskatchewan 
Conservative Markov operators from 1D free harmonic analysis
The reciprocal Cauchy transform of a distribution has played
an important role in the scalarvalued free probability, especially
in the calculation of free convolution of measures. In this talk, we
will discuss how methods and techniques of onedimensional free harmonic
analysis lead to a new class of conservative dynamical systems, in which
the underlying state space is an infinite measure space. This extends
the old results of Aaronson on the ergodic theory for inner functions
on the complex upper halfplane.

Participants as of June 27, 2013
Full Name 
University/Affiliation 
Anderson, Greg 
University of Minnesota 
Andrews, Rob 
(no affiliation) 
Arizmendi Echegaray, Octavio 
Universität des Saarlandes 
Barmherzig, David 
University of Toronto 
Basu, Madhushree 
Institute of Mathematical Sciences 
Belinschi, Serban 
Queen's University 
Ben Hamza, Abdessamad 
Concordia University 
Biane, Philippe 
Universite ParisEst (MarnelaVallee) 
Blitvic, Natasha 
Vanderbilt Univ 
Capitaine, Mireille 
Université Paul Sabatier 
Cheng, Oliver 
Brown University 
Dabrowski, Yoann 
Université Lyon 1 
Dewji, Rian 
University of Cambridge 
Edelman, Alan 
Massachusetts Institute of Technology 
Ejsmont, Wiktor 
University of Wroclaw 
Elliott, George 
University of Toronto 
Farah, Ilijas 
York University 
Farrell, Brendan 
California Institute of Technology 
Fevrier, Maxime 
Université ParisSud 11 
Friedrich, Roland 
HumboldtUniversität zu Berlin 
Gerhold, Malte 
Universität Greifswald 
Grabarnik, Genady 
St John's University 
Grela, Jacek 
Jagiellonian University 
Gu, Yinzheng 
Queen's University 
Guionnet, Alice 
Massachusetts Institute of Technology 
Halevy, Itamar 
no affiliation 
Hayes, Benjamin 
University of California, Los Angeles 
Hu, Yi 
Colorado State University 
Huang, HaoWei 
Indiana University, Bloominton 
Huang, Xuancheng 
University of Toronto 
Jain, Madhu 
Indian Institute of Technology, Roorkee 
Jeong, Ja A 
Seoul National University 
Kappil, Sumesh 
Indian Statistical Institute 
Kennedy, Matthew 
Carleton University 
Koestler, Claus 
University College Cork 
Kuan, Jeffrey 
Harvard University 
Lee, Eunghyun 
Université de Montréal 
Lenczewski, Romuald 
Wroc?aw University of Technology 
Li, Boyu 
University of Waterloo 
Liu, Weihua 
University of California 
Male, Camille 
Université ParisDiderot 
Mastnak, Mitja 
Saint Mary's University 
Merberg, Adam 
University of California, Berkeley 
Miller, Tomasz 
Warsaw University of Technology 
Mingo, James A. 
Queen's University 
Movassagh, Ramis 
Northeastern University 
Muraki, Naofumi 
Iwate Prefectural University 
Nica, Alexandru 
University of Waterloo 
Nica, Mihai 
Courant Institute of Mathematical Sciences 
Novak, Jonathan 
Massachusetts Institute of Technology 
Nowak, Maciej 
Jagiellonian University 
Pachl, Jan 
Fields Institute 
PérezAbreu, Víctor 
Center for Research in Mathematics 
Redelmeier, Emily 
Université ParisSud XI 
Shlyakhtenko, Dimitri 
University of California, Los Angeles 
Skoufranis, Paul 
University of California, Los Angeles 
Speicher, Roland 
Saarland University 
Spektor, Susanna 
University of Alberta 
Sunder, Viakalathur S. 
The Institute of Mathematical Sciences 
Szpojankowski, Kamil 
Warsaw University of Technology 
Tarrago, Pierre 
University ParisEst (Marne la Vallee) 
Vargas, Carlos 
Universität des Saarlandes 
Viola, Maria Grazia 
Lakehead UniversityOrillia 
Voiculescu, DanVirgil 
University of California, Berkeley 
Wang, JiunChau 
University of Saskatchewan 
Warchol, Piotr 
Jagiellonian University 
Weber, Moritz 
Saarland University 
Williams, John 
Texas A&M University 
Zhou, Youzhou 
McMaster University 
Back to main index
Back to top

