SCIENTIFIC PROGRAMS AND ACTIVITIES

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

The spectrum of permutation invariant matrices

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.