SCIENTIFIC PROGRAMS AND ACTIVITIES

October 23, 2014

THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

July 2013
Focus Program on Noncommutative Distributions in Free Probability Theory

July 2-6, 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)


Tuesday July 2
8:45 - 9:15
On-site Registration
9:15 - 9:30
Welcome and Introduction
9:30 - 10:30
Dan-Virgil 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 Paris-Diderot (slides)
The spectrum of permutation invariant matrices
4:00 - 4:30
Maxime Fevrier, Université Paris-Sud 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 Paris-Est (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
Jiun-Chau Wang, University of Saskatchewan
Conservative Markov operators from 1-D 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 operator-valued 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 q-deformation of free independence
4:00 - 4:30
Emily Redelmeier, Université Paris-Sud XI
Quaternionic Second-Order Freeness
4:30 - 5:00
Madhushree Basu, Institute of Mathematical Sciences
Continuous Courant-Fischer-Weyl minimax theorem
Saturday, July 6
9:30 - 10:30
Mireille Capitaine, Université Paul Sabatier (slides)
Exact separation phenomenon for the eigenvalues of large Information-Plus-Noise 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 (not-very-)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 down-to-earth and explicit fashion.

Basu, Madhushree
Institute of Mathematical Sciences

Continuous Courant-Fischer-Weyl minimax theorem

A minimax theorem is a result that gives a characterization of eigenvalues of compact self-adjoint operators on Hilbert spaces. The Courant-Fischer-Weyl 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 self-adjoint 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 self-adjoint 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 operator-valued 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 non-selfadjoint 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 Paris-Est

From noncrossing partitions to ASM

Noncrossing partitions are related to alternating sign matrices through the Razumov-Stroganov (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 Information-Plus-Noise type matrices. Application to spiked models
We consider large Information-Plus-Noise 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 Marchenko-Pastur distribution with parameter c by x${\mapsto \sqrt{x}}$. We use the previous results to characterize the outliers of spiked Information-Plus-Noise type models.
Alan Edelman
Massachusetts Institute of Technology

Isotropic Entanglement: A Fourth Moment Interpolation Between Free and Classical Probability

The difference between the non-commutative 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 Haar-distributed 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 Paris-Sud 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 alpha-stable entries. This talk is based on joint works with Bordenave, Benaych-Georges 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 Paris-Diderot

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 many-body (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 non-commuting 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 q-deformation of free independence

I will introduce a product operation (q-product) for non-commutative probability spaces. There exists naturally a notion of `q-independence' which is associated to the q-product. Here an independence is understood as a universal calculation rule for mixed moments of non-commutative 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 q-product can be proved based on the card arrangement. The q-deformed cumulants is also discussed.
Nica, Alexandru
University of Waterloo

Some remarks on the combinatorics of free unitary Brownian motion

It is well-known 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 Guay-Paquet.
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 finite-dimensional or almost finite-dimensional 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 non-linear 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 Burgers-like 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 Paris-Sud XI Quaternionic Second-Order Freeness
Second-order 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 second-order 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 non-atomic spectral measures, then ${Y=p(X_1,\dots,X_n)}$ has a non-atomic spectral measure, for any non-constant (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 operator-valued 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 non-selfadjoint case will be presented in the talk of Belinschi.
Carlos Vargas
Universität des Saarlandes

Block modifications of the Wishart ensemble and operator-valued free multiplicative convolution

We describe block modifications of $nd \times nd$ Wishart matrices: For a self-adjoint 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 operator-valued free probabilistic approach. It turns out that \mu^{\varphi} is exactly the matrix-valued 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, Dan-Virgil
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 bi-freeness.
Wang, Jiun-Chau
University of Saskatchewan

Conservative Markov operators from 1-D free harmonic analysis

The reciprocal Cauchy transform of a distribution has played an important role in the scalar-valued free probability, especially in the calculation of free convolution of measures. In this talk, we will discuss how methods and techniques of one-dimensional 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 half-plane.


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 Paris-Est (Marne-la-Vallee)
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 Paris-Sud 11
Friedrich, Roland Humboldt-Universität 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, Hao-Wei 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 Paris-Diderot
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
Pérez-Abreu, Víctor Center for Research in Mathematics
Redelmeier, Emily Université Paris-Sud 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 Paris-Est (Marne la Vallee)
Vargas, Carlos Universität des Saarlandes
Viola, Maria Grazia Lakehead University-Orillia
Voiculescu, Dan-Virgil University of California, Berkeley
Wang, Jiun-Chau 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