# Free probability and the large N limit, III: Abstracts

**Stephen Avsec**

*A Characterization of Noncommutative Brownian Motion*

**Abstract:**We prove an equivalence between a class of noncommutative Brownian motions and real-valued positive definite functions with a special invariance property on the infinite symmetric group. Our invariance property ensures that the Brownian motion is represented in a Type II

_{1}von Neumann algebra. Our construction generalizes theconstruction of

*q*-Brownian motion of Bozejko and Speicher and is related to the approach of Guta and Maassen. In certain cases, we are able to establish the weak* CBAP for these algebras. This is joint work with Marius Junge.

**Sebran Belinschi**

*L*

^{∞}-bound for densities of free convolutions, unitary integrals and subordination**Abstract:**It is known that free additive convolution has a very strong regularizing effect. In this talk, we will discuss a stronger result, stating roughly that the density of the free additive convolution of two nontrivial compactly supported probability measures is almost always bounded. Following recent work of Kargin, this result has certain applications in the analysis of deformed unitarily invariant matrix models. We will discuss some of these applications and indicate connections between Biane's subordination functions and some unitarily invariant random matrices.

Hari Bercovici

Hari Bercovici

*R-diagonal convolution on the positive half-axis (analytic aspects)*

**Abstract:**Let

*D*

^{+}denote the space of probability measures on [0, ∞). I will present an ongoing joint work with Andu Nica and Michael Noyes, concerning a convolution operation on

*D*

^{+}which originates from the operation of ⊞-convolution for

*R*-diagonal elements. My talk will focus on analytic aspects of this ``

*R*-diagonal convolution'', and is coordinated with the talk of Andu Nica, who will discuss combinatorial aspects (in the framework of compact support).

**Natasha Blitvic**

*The (q,t)-Gaussian Process*

**Abstract:**According to Speicher's Non-commutative Central Limit Theorem, stochastic mixtures of commuting and anti-commuting elements provide asymptotic models for generalized Gaussian, namely q-Gaussian, statistics. The same statistics can also be seen to arise in the context of quantum harmonic oscillators and have representations on a twisted Fock space. In this talk, I will describe a generalization of Speicher's result with the commutation/anti-commutation requirement lifted to admit a broader commutation structure. The resulting generalization leads to a second-parameter refinement of the q-Gaussian statistics and this refinement, in turn, gives rise to a highly natural two-parameter deformation of the classical Fock spaces. Time permitting, we will also overview the associated random matrix models and highlight a specialization that yields an interesting deformation of free probability.

Stephen Curran

Stephen Curran

*On the quantum double of a finite depth planar algebra*

**Abstract:**Ocneanu’s asymptotic inclusion for finite index, finite depth inclusions of hyperfinite type II

_{1}factors can be viewed as a subfactor analogue of Drinfeld's quantum double construction for finite-dimensional Hopf algebras. Popa's symmetric enveloping inclusion generalizes this construction to arbitrary finite index inclusions of II

_{1}factors, and encodes a number of important analytic properties of the original subfactor. In recent joint work with V. Jones and D. Shlyakhtenko, we have given a diagrammatic description of the symmetric enveloping inclusion associated to the planar algebra subfactors of Guionnet-Jones-Shlyakhtenko. In this talk we explain how this construction can be used to compute the planar algebra of the asymptotic inclusion, in the finite-depth case. As an application we give a planar algebraic computation of the fusion rules for bimodules arising from the asymptotic inclusion, recovering in this way some well-known results of Ocneanu and Evans-Kawahigashi.

**Amir Dembo**

*Low temperature expansion for matrix models*

**Abstract:**Relying on its representation as a solution of certain Schwinger-Dyson equation, we study the low temperature expansion of the limiting spectral measure (and limiting free energy), for random matrix models, in case of potentials which are strictly convex in some neighborhood of each of their finitely many local minima. When applied to suitable polynomial test functions, these expansions are given in terms of the aboslutely convergent generating function of an interesting class of colored maps.

This talk is based on a joint work with Alice Guionnet and Edouard Maurel-Segala.

**Ken Dykema**

*Matricial microstates and sofic approximations for amalgamated free products.*

**Abstract:**We recall Voiculescu's notion of matricial microstate and some results about these for amalgamated free products. Analogous considerations apply for sofic approximations of groups, including the sofic dimension of groups, which is like free dimension in the context of von Neumann algebras. Finally, we discuss some ideas about trying to find a non-sofic group.

**Steven Evans**

*Eigenvalues of large random trees*

**Abstract:**A common question in evolutionary biology is whether evolutionary processes leave some sort of signature in the shape of the phylogenetic tree of a collection of present day species. Similarly, computer scientists wonder if the current structure of a network that has grown over time reveals something about the dynamics of that growth. Motivated by such questions, it is natural to seek to construct ``statistics'' that somehow summarize the shape of trees and more general graphs, and to determine the behavior of these quantities when the graphs are generated by specific mechanisms.

The eigenvalues of the adjacency and Laplacian matrices of a graph are obvious candidates for such descriptors. I will discuss how relatively simple techniques from linear algebra and probability may be used to understand the eigenvalues of a very broad class of large random trees. These methods differ somewhat from those that have been used thusf ar to study other classes of large random matrices.

This is joint work with Shankar Bhamidi (U. of North Carolina) and Arnab Sen (University of Cambridge).

**Maxime Fevrier**

*First and higher order infinitesimal freeness*

**Abstract:**We define a natural generalization of the notion of infinitesimal freeness, first introduced by Belinschi and Shlyakhtenko. It is devoted to the local study (when the parameter goes to zero) of the free product of two families of noncommutative distributions indexed by some set having 0 as an accumulation point. We give a quite simple and general framework for the combinatorial study of infinitesimal freeness. Among the perspectives of applications, there is the global behaviour of some large random matrices. The first order part of the talk is based on joint work with A. Nica.

**Alberto Grunbaum**

*Classical and quantum walks: any connections to free probability???*

**Abstract:**I will talk mainly about quantum walks, recurrence, limit laws (analogs of the central limit theorem) and will show lots of pictures. In the classical case, I learned by talking to Michael Anshelevich of remarkable connections between free probability and classical work of Karlin and McGregor. Can sone of this connections also exist in the quantum case??

**Michael Hartglass**

*The GJS Construction in Infinite Depth*

**Abstract:**In 1995, Popa introduced λ-lattices and constructed a tower of II

_{1}factors

*M*

_{0}⊂

*M*

_{1}⊂ ... ⊂

*M*

_{n}⊂ ... realizing a given λ-lattice as its standard invariant. In 2009, Guionnet, Jones, and Shlyakhtenko provided a planar algebra construction for a tower of II

_{1}factors with a given standard invariant

*P*. They proved that in the case that

*P*is finite depth, the factor

*M*is isomorphic to

_{k}*L*(

**F**(1 + δ−2

*k*(δ − 1)

*I*)) where δ and

*I*are the loop parameter and global index of

*P*respectively. In this talk, I will use standard embedding techniques of Dykema and Redelmeier to show that when

*P*is infinite-depth, the factors in the tower are all isomorphic to L(

**F**(∞)).

**Benjamin Hayes**

*An l*

^{p}-Version of von Neumann Dimension for Banach Space Representations of Sofic Groups**Abstract:**In 1996, Voiculescu discovered a formula for von Neumann dimension for amenable groups analogous to the entropy of a dynamical system. In 2010, Antoine Gournay developed a notion of

*l*-dimension for representations of an amenable group

^{p}*G*which are contained in a finite direct sum of the left regular represntation on l

*. We define a notion of*

^{p}*l*-Dimension for sofic groups

^{p}*G*and use this to show that (

*l*)

^{p}^{⊕n}are non-isomorphic for different values of

*n*. The same result also holds if

*G*satisfies the Connes Embedding Conjecture.

**Todd Kemp**

*Liberating Projections*

**Abstract:**The free liberation process (essentially conjugation by a free unitary Brownian motion) was introduced by Voiculescu in the study of free entropy. It is used in the definition of χ*, and so understanding its regularity is key to approaching some important outstanding problems in free probability.

In this talk, I will discuss joint work with Benoit Collins on the free liberation process in the von Neumann algebra generated by two projections. The operator-valued angle between them, which characterizes the von Neumann algebra, can be studied through the Cauchy transform of its law; this Cauchy transform satisfies a semi-linear holomorphic PDE. Using a mixture of classical PDE theory and an adaptation of Biane's subordination technology, we show that the flow has very strong smoothing properties. The resulting regularity theory, in turn, has significant applications to the free entropy.

**Franz Lehner**

*On Characterization Problems in Free Probability*

**Abstract:**We present characterizations of the semicircle law by freeness of linear and quadratic forms without boundedness assumptions.

Camille Male

Camille Male

*Distributions of traffics of large random matrices and their free product.*

**Abstract**: Motivated by natural question in random matrix and free probability theory, we introduce an abstract notion of traffics. It should be applied for families of larges random matrices, finitely generated random groups and infinite random rooted networks with uniformly bounded degree. We introduce a notion of freeness of traffics, which contains both the classical notion of independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem for families of matrices invariant by permutation, which enlarges the class of large random matrices for which we can predict the empirical eigenvalues distribution. We prove a central limit for the sum of free traffics, and interpret the limit in law as the (traffic)-convolution of a gaussian commutative random variable and a semicircular non commutative random variable. We make a connection between the freeness of traffics and the natural free product of random graphs, the combination of the statistical independence and the geometric free product. We give perspective for a problem of Lovász about defining algebras of linear combinations of graphs with appropriate convergence properties, in the scale of Benjamini-Schramm-Aldous-Lyons's weak local convergence regime.

**Adam Merberg**

*A continuous deformed free group factor*

**Abstract:**We discuss a Fock space associated to a symmetric function Q on U

^{2}taking values in (-1,1), where U is a nonempty open subset of R

^{j}for some j. The creation and annihilation operators on this space will give rise to operator-valued distributions satisfying a Q-deformed commutation relation. Analogous to the q

_{ij}-Fock space of Bozejko and Speicher, we have field operators arising as the sum of the creation and annihilation operators. Our main result is that these operators generate a factor which does not have property gamma.

**James Mingo**

*Second Order Even and R-diagonal Operators*

**Abstract:**Voiculescu showed that circular operators, the free analogue of complex Gaussian random variables, and Haar unitaries are related by polar decomposition. This relationship was deepened by Nica and Speicher who showed that both are examples of R-diagonal operators. Nica and Speicher also showed that R-diagonal operators and even operators, self-adjoint operators with vanishing odd moments, are related algebraically and via their free cumulants.

A few years ago in joint work with Collins, Sniady, and Speicher, we came up with the idea of higher order freeness. In particular second order freeness allows one to analyse the fluctuations of the sum or product of random matrix ensembles if they are free of second order. We extend the results of Nica and Speicher to the case of second order freeness.

This is joint work with Octavio Arizmendi.

**Alexandru Nica**

*R-diagonal convolution on the positive half-axis (combinatorial aspects)*

**Let**

Abstract:

Abstract:

*D*

_{c}

^{+}denote the space of probability measures with compact support on [0, ∞). I will present an ongoing joint work with Hari Bercovici and Michael Noyes, concerning a convolution operation on

*D*

_{c}

^{+}which originates from the operation of ⊞-convolution for

*R*-diagonal elements. My talk will focus on combinatorial aspects of this ``

*R*-diagonal convolution'', and is coordinated with the talk of Hari Bercovici, who will discuss analytic aspects (not necessarily limited to the framework of compact support).

**Jesse Peterson**

*Closable derivations and unique group-measure space decomposition*

**Abstract:**I will show how the theory of closable derivations can be applied to von Neumann algebras to give examples of II

_{1}factors with unique group-measure space Cartan subalgebra. In the process, I will also show that for every group with positive, finite first

*l*

^{2}-Betti number, there exists an ergodic action such that the corresponding group-measure space construction has trivial fundamental group.

**David Renfrew**

*Outliers in the Spectrum of Finite Rank Deformations to Wigner Random Matrices*

**Abstract:**We consider eigenvalues of finite rank deformation to Wigner matrices that lie outside of the support of the semicircle. It was first shown by M.Capitaine, C. Donati-Martin, and D. Feral that the fluctuations of these eigenvalues are non-universal. We extend these results to the optimal class of random matrices and more general perturbations.

**Dima Shlyakhtenko**

*Free Monotone Transport*

**Abstract:**(joint work with A. Guionnet) Using a free analog of the Monge-Ampere equation, we show existence of monotone transport for free Gibbs states that are small perturbations of the semicircular law. We discuss this result and its applications.

**Gabriel Tucci**

*Some Results on Random Vandermonde Matrices*

**Abstract:**We present some results in connection with random Vandermonde matrices. An

*N*×

*N*matrix

**V**with unit complex entries is a

*random Vandermonde matrix*if there exist random variables θ

_{1},...,θ

_{N},

*N*∈[0,1] such that

*V*

_{N}(

*p*,

*q*):=1/√

*N*

*x*

_{q}^{p-1}where

*x*

_{q}:=

*e*

^{2π i θq}. We assume that the random variables are i.i.d. in the unit interval. Upper and lower bounds for the maximum singular value are found to be

*O*(log

*N*) and

*O*(log

*N*/log log

*N*). We further study the behavior of the minimum singular value. In particular, we prove that the minimum singular value is at most

*N*

^{2}exp(-

*C*√

*N*) where

*C*is a constant independent on

*N*. Lastly, for each sequence of positive integers {

*k*

_{p}}

_{p=1}

^{∞}we present a generalized version of the previously discussed matrices. The classical random Vandermonde matrix corresponds to the sequence

*k*

_{p}=

*p*-1. We find a combinatorial formula for their moments and show that the limit eigenvalue distribution converges to a probability measure supported on [0,∞). Finally, we show that for the sequence

*k*

_{p}=2

^{p}the limit eigenvalue distribution is the famous Marchenko-Pastur distribution.

**Dan Voiculescu**

*Is there a mod Hilbert-Schmidt BDF-type Theorem for Operators with Trace-Class Self-Commutator?*

**John D. Williams**

*A Hincin Type Characterization of Infinite Divisibility for Operator Valued Free Probability*

**Abstract:**Operator valued free probability theory was developed in the 1990's as a method of encoding amalgamated free product phenomenon in operator algebras in a probabilistic setting. As in the case of scalar valued free probability, this theory may be developed along similar lines as classical probability theory, with many classical theorems having free analogues. In classical probability theory, it was proven by Hincin that a probability measure that is the weak limit of the convolution of an infinitesimal array of probability measures is necessarily infinitely divisible. We will provide an alternative proof of this theorem utilizing the Steinitz Lemma. We will then use this approach to prove an analogous result in the field of operator valued free probability theory.

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