Schedule for: 16w5025 - Analytic versus Combinatorial in Free Probability

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The talk will be about extreme values in free probability and the beginning of an extension to bi-free probability, that is to free probability with left and right variables.

Bi-free probability is a generalization of free probability to study pairs of left and right faces in a non-commutative probability space. In this talk, I will demonstrate a characterization of bi-free independence inspired by the ``vanishing of alternating centred moments'' condition from free probability. I will also show how these ideas can be used to introduce a bi-free unitary Brownian motion and a liberation process which asymptotically creates bi-free independence.

We will discuss new results and open questions in bi-free harmonic analysis, in which the left and right variables are assumed to commute with each other. The topics include limit theorems and the bi-free Levy-Khintchine formulas for multiplicative and additive bi-free convolutions. These are joint with Hao-Wei Huang and Takahiro Hasebe.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

We consider the Boolean version of Voiculescu's extension from free probability to bi-free probability and introduce the notion of bi-Boolean independence for non-unital pairs of algebras. We show that both the combinatorial and the analytic aspects of bi-free probability have immediate analogues with respect to this new notion of independence and discuss how to extend it further to the operator-valued setting. (Joint work with Paul Skoufranis.)

The talk is based on two works:  one  with S. Belinschi  and  another one with S. Belinschi and H. Bercovici. I will present the main  ideas we use to prove: that,  a.s. for large dimension, there is no eigenvalue in any interval lying outside the support of some deterministic probability measure which is computed  with the tools of free probability; the strong asymptotic freeness of independent Wigner matrices and any family of deterministic matrices with strong limiting distribution; the location of outliers of a polynomial in a Wigner matrix and a spiked deterministic matrix.

Moving beyond the Wigner matrix paradigm, there is a vast literature on "band matrices" -- random matrices with entries that are not identically distributed.  In almost all cases, however, it is still assumed that the entries are independent (modulo the Hermitian condition), in order to prove convergence of the empirical eigenvalue distribution. One case where independence has been relaxed is block matrices.  If $X_n$ is a random Hermitian matrix with blocks of size $m\times m$ that are independent, then operator valued free probability may be used to analyze the limit empirical spectral distribution (as discovered by Shlyakhtenko, and continued in the work of Bryc, Oraby, Rashidi Far and Speicher).  This analysis, so far, can only handle the case that $m$ is constant, or, dually, $m$ is proportional to $n$. In this talk, I will discuss recent joint work with my former student David Zimmermann, where we handle a much more general intermediate case.  We show that, if the matrix $X_n$ can be partitioned into independent ``blocks'' (not necessarily rectangular) each of size $o(\log n)$, then the empirical spectral distribution converges to its mean in probability.  (The limit is usually not semicircular.)  The main tool we use is a mollified logarithmic Sobolev inequality, which I will discuss.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

My talk will discuss ongoing work in developing finite random matrix techniques that have been motivated by free probability.  The key player on the finite side of the correspondence will be the characteristic polynomial.  I will show that certain binary operations on polynomials converge to the free additive and multiplicative convolutions and how one can use these operations to make statements about discrete, finite matrix ensembles that exhibit "free-like" behavior.  In particular, I will introduce a free probability analogue of the "Poisson paradigm," as introduced by Alon and Spencer in their book on Probabilistic Combinatorics.

  Let $A_1,\dots,A_L$ be adjacency matrices of independent random graphs $G_1,\dots,G_L$ on the vertex set $\{1,...,N\}$. Assume for each $\ell=1,\dots,L$ that the expected degree of a vertex of $G_\ell$ uniformly chosen at random goes to infinity, and that each graph is invariant in law by relabelling of its vertices. We state a quantitative estimate of decorrelation on the edges of the graphs that implies the asymptotic freeness of well-normalized versions of $A_1,\dots,A_L$, as well as their asymptotic freeness with arbitrary matrices. We prove that this estimate holds for the uniform simple $d_N$-regular graph with $|d_N-\frac N 2|$ going to infinity fast enough. The proof is based on asymptotic traffic independence and combinatorial manipulations.

For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, Cébron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau)$ that extends the trace $\varphi$ [CDM16]. This construction comes equipped with some canonical independence structure: in a joint work in progress with Male, we show that $(\mathcal{G}(\mathcal{A}), \tau)$ can be realized as the free product of three natural subalgebras (in the sense of Voiculescu), and that there exists a canonical homomorphic conditional expectation $P$ onto a subalgebra intermediate to $\mathcal{A}$ and $\mathcal{G}(\mathcal{A})$. Combining this with the coherent convergence properties of $(\mathcal{G}(\mathcal{A}), \tau)$ proved in [CDM16], we show that free independence describes the asymptotic behaviour of a large class of dependent random matrices (in particular, we recover and explain a result of Bryc, Dembo, and Jiang on random Markov matrices [BDJ06]).

Biane-Perelemov-Popov matrices are a family of quantum random matrices which quantize uniformly random Hermitian matrices with deterministic eigenvalues and uniformly random eigenvectors.  BPP matrices depend on a semiclassical parameter which controls the degree of noncommutativity of their entries.  Improving on results of Biane and Collins-Sniady, and resolving a conjecture of Bufetov-Gorin, we demonstrate that classically independent BPP matrices become asymptotically free provided the semiclassical parameter decays as the dimension increases.  This result has consequences in asymptotic representation theory, implying in particular that $GL_N(\mathbb{C})$ tensor products are described by free probability in any and all semiclassical limits.  This is joint work with Collins and Sniady.

In this talk we will explain recent results with Daniel Perales where we define cumulants for finite free convolution. We give a moment-cumulant formula and show that these cumulants satisfy desired properties: they are additive with respect to finite free convolution and they approach free cumulants as the dimension goes to infinity.

This talk focuses on some limit theorem for positive multiplicative free Levy processes at time near 0. The limit distributions are "log free stable laws", which seem to be universal. This work is related to Tucci-Haagerup-Moeller's multiplicative free analogue of the law of large numbers, where they investigated a limit theorem for large time. This research is a joint work with Octavio Arizmendi.

About a decade ago, Mingo, Nica, and Speicher studied the fluctuations of the moments of Gaussian matrices from a combinatorial perspective. Based on their work, in this talk we will study the fluctuations of the moments of block Gaussian matrices. In particular, we will find a semi-explicit formula for a matricial version of the second-order Cauchy transform. Using the linearization technique, this formula will provide the second-order Cauchy transform of polynomials in Gaussian matrices. This is joint work with Serban Belinschi and James Mingo.

I will give a general introduction to compact matrix quantum groups and then focus in a survey talk on those of combinatorial type. These include Banica-Speicher's orthogonal easy quantum groups using partitions; the unitary easy quantum groups introduced by Pierre Tarrago and myself using partitions with monochromatic colored points; the spatial partition quantum groups introduced by Guillaume Cébron and myself using three-dimensional partitions; and finally the partial commutative quantum groups introduced by Roland Speicher and myself fitting to the mixtures of classical and free independence by Mlotkowski and Speicher-Wysoczanski.

Motivated by recent advances on mixtures of classical and free probabilities, we show that the vanishing of certain cumulants implies epsilon-independence. Joint work with K. Ebrahimi-Fard and R. Speicher.

In this talk we'll survey the machinery of half-shuffles (co)products. It will then be connected to the construction of free cumulants. Related group theoretical aspects will be mentioned. This talk is based on joint work with F. Patras (CNRS, Nice, France).

In this talk we briefly review the appearance of Lie groups and Lie algebras in non-commutative probability theory and their roles as moment and cumulant co-ordinate spaces, respectively. We shall discuss a general structure theorem and illustrate it with an example.

We show that the von Neumann algebra of a finitely generated finitely presented sofic group is strongly 1-bounded in the sense of Jung. As a corollary we obtain that such a von Neumann algebra cannot be written as a non-trivial free product and in particular cannot be a free group factor.

In their 2015 paper, Ledoux, Nourdin, and Peccati used Stein kernels and Stein discrepancies to improve the classical logarithmic Sobolev inequality (relative to a Gaussian distribution). Simply put, Stein discrepancy measures how far a probability distribution is from the Gaussian distribution by looking at how badly it violates the integration by parts formula. In free probability, free semicircular operators are known to satisfy a corresponding “integration by parts formula” by way of the free difference quotients. Using this fact, we define the non-commutative analogues of Stein kernels and Stein discrepancies and use them to produce an improvement of Biane and Speicher’s free logarithmic Sobolev inequality from 2001. We will also see several examples of free Stein kernels which have interesting connections to free monotone transport.

In this talk, we will provide an overview of the structures and constructions developed to study bi-free independence. In particular, we will focus on several results pertaining to operator-valued bi-free probability. These results are important in developing a rich theory to apply to the scalar-valued setting and giving a true sense of what bi-free probability is. In particular, by studying operator-valued bi-free probability one obtains a good notion of the analogue of random matrix models in the bi-free setting.

In classical probability there are known many characterization of probability measures by independence properties. The best example of such result is Bernstein's theorem, which says that for independent X and Y, random variables X+Y and X-Y are independent if and only if X and Y have Gaussian distribution. Surprisingly this theorem and many other characterizations have their counterparts in free probability. My talk will be devoted to present two techniques to deal with characterizations problems in free probability: one combinatorial and one analytic (which uses subordination functions of free additive and multiplicative convolutions). I will also present known analogies between characterization problems in classical and free probability. Talk is based on joint work with Wiktor Ejsmont (Wroclaw) and Uwe Franz (Besancon).

We show that there exists several reduced-like representations of a full free product C*-algebra  that gives natural   reduced-like free products. These algebras are useful for example when one wants to understands free products with respect to non GNS faithfull conditional expectations.  At the same time, they share the basic properties of the classical reduced free product (stability of exactness, K-theory computation,...).  We will also explain a result about the continuity of some radial  maps.

In the early 2000's, Fumio Hiai introduced the $q$-Araki Woods von Neumann algebras, which are generalizations of both Shlyakhtenko's free Araki-Woods factors and Bozejko-Speicher's $q$-Gaussian algebras.  In comparison to the free Araki-Woods and the $q$-Gaussian cases, very little is known about the structure of generic $q$-Araki Woods algebras.  In particular, their type classification, factoriality, (strong) solidity, and even injectivity is not fully understood.  The reason for this lack of progress can be chalked up to the fact that when dealing with $q$-Araki-Woods algebras, one simultaneously looses the ''nice'' properties of free independence and traciality.  In this talk, I will discuss the problem of computing the cb-norms of an important class of linear maps on $q$-Araki-Woods algebras called radial multipliers.  Although we are unable to obtain nice formulas for the cb norms of radial multipliers (as was achieved in the free case by Houdayer-Ricard), we are able to use non-commutative central limit and ultraproduct techniques to show that their cb-norms do not depend on the underlying orthogonal transformation group governing the non-traciality of the algebra.  As a consequence of this deformation-invariance property for radial multipliers, we establish that all $q$-Araki-Woods algebras have the complete metric approximation property (CMAP).  This talk is based on joint work with Mateusz Wasilewski (IMPAN).

We start from recalling the generalization of the R-transform for strictly-nonhermitian large random matrices. Then we present the generalization of the Voiculescu equation for nonhermitian operators, pointing at the entangled coevolution of the spectra and eigenvectors. We exemplify this evolution solving the simplest, Gaussian case (Ginibre ensemble). Finally, basing on the recent work by S. Belinschi, M.A. Nowak, R. Speicher and W. Tarnowski, we show how to augment the single ring theorem (Haagerup-Larsen theorem) with the predictions for certain left-right eigenvector correlations. We conclude with presentation of few examples (products/ratios of Ginibre ensembles, arbitrary product of truncated unitary matrices , sum of arbitrary number of unitary ensembles).

We investigate various aspects of quadratic forms in free random variables: Characterization problems, distributions and preservation of infinite divisibility. A crucial property in these respects is the phenomenon of cancellation of odd cumulants and we give a characterization of quadratic forms which exhibit this phenomenon.

The classical Bernstein inequality shows that the sum of bounded independent random variables is concentrated around its expectation in terms of the variance of the sum’s increments. In this talk, we are interested by a Bernstein-type inequality for the largest eigenvalue of the sum of dependant Hermitian random matrices with bounded operator norm. In the case where the matrices are independent, the matrix version of the Bernstein inequality was due to Ahlswede et Winter (2002) and was then improved by Tropp (2012). In this talk, we shall see how to relax the independence condition and extend this inequality to a class of dependent matrices. We shall also shed light on some difficulties arising from the non-commutative and dependence setting. (joint work with F. Merlevède and P. Youssef)

We study the eigenvalues of polynomials of large random matrices which have only discrete spectra. Our model is closely related to and motivated by spiked random matrices, and in particular to a recent result of Shlyakhtenko in which asymptotic infinitesimal freeness is proved for rotationally invariant random matrices and finite rank matrices. We show the almost sure convergence of Shlyakhtenko's result. Then we show the almost sure convergence of eigenvalues of our model when it has a purely discrete spectrum. We define a framework for analyzing purely discrete spectra and develop the moment method for the eigenvalues of compact (and in particular Schatten class) operators. We give several explicit calculations of purely discrete eigenvalues of our model. This talk is based on a joint work with B. Collins and T. Hasebe.

We show that a sequence of random Vandermonde matrices based on i.i.d entries on the unit circle has asymptotic *-distribution, which is that of a B-valued R-diagonal elements, where B is the algebra $C[0,1]$.  (Joint work with March Boedihardjo.)

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.