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

**Michael Anshelevich,**

**Free convolution semigroups and coefficient stripping**

**.**

Abstract.Coefficient stripping is a map from probability measures with finite variance to probability measures. We will see that additive free convolution semigroups behave well with respect to this map. Namely, applying coefficient stripping to a free convolution semigroup always produces semicircular evolution. On the other hand, applying the inverse of this map to a freely evolving family always produces a two-state free convolution semigroup. A number of examples will be provided.

Abstract.

**Octavio Arizmendi,**

*K***-divisible partitions in first and second order freeness.**

Abstract.

Abstract.

*K*-divisible non-crossing partitions partitions appear when trying to understand the combinatorics of moments and cumulants of products of k free random variables. A few years ago, Collins, Mingo, Sniady, and Speicher, gave a notion of higher order freeness. In particular second order freeness allows to analyse the fluctuations of the sum or product of random matrix ensembles if they are free of second order. As we well explain, it turns out that similarly as for the first order case the combinatorics of moments and cumulants of products of second free random variables are governed by

*k*-divisible non-crossing partitions on an annulus. We will apply these formulas to

*R*-diagonal operators in the second order level. A prominent example which comes from random matrices is that of products of circular operators. This is work in progress with James Mingo.

**Stephen Avsec, Recent results on Noncommutative Exchangeable Brownian Motions.**

Abstract.We will begin with a recent definition of noncommutative brownian motion due to B. Collins and M. Junge. We will then discuss recent progress toward characterizing these noncommutative brownian motions in the exchangeable case using operator-valued random variables. This includes joint work with Benoit Collins and Marius Junge.

Abstract.

**Serban Belinschi, On the free convolution with an operator-valued semicircular: an exercise following Biane**

Abstract.Based on ideas from recent work with Mai and Speicher and older work with Anshelevich, Fevrier and Nica, in this talk we will discuss to what extent the analytic methods employed by Biane in his 1997 work "On the free convolution with a semi-circular distribution" apply to the operator valued case. While there are several commonalities, it turns out that differences, both in the nature of the tools and the nature of the results are significant. This is part of work in progress.

Abstract.

**Natasha Blitvic, SIF Permutations and Chord-connected Permutations.**

Abstract.Stabilized-interval-free (SIF) permutations, introduced by Callan in 2004, have recently appeared in the work of Ardila, Rincon, and Williams in connection to positroid enumeration. Such permutations allow for a factorization of permutations along non-crossing partitions and, as such, have a probabilistic interpretation in terms of the free moment-cumulant formula. In this talk, I will revisit an older (2009) result in which SIF permutations can in turn be factored into ``chord-connected permutations" and discuss some related combinatorial results and probabilistic interpretations.

Abstract.

**Charles Bordenave,**

**Outlier eigenvalues for deformed i.i.d. random matrices**

**.**

Abstract.This is joint work with Mireille Capitaine. We consider a square random matrix of size

Abstract.

*N*of the form

*A*+

*Y*where

*A*is deterministic and

*Y*has iid entries with variance 1/

*N*. Under mild assumptions, as

*N*grows, the empirical distribution of the eigenvalues of

*A*+

*Y*converges weakly to a limit probability measure β on the

complex plane. This work is devoted to the study of the outlier eigenvalues, i.e. eigenvalues in the complement of the support of β. Even in the simplest cases, a variety of interesting phenomena can occur. As in earlier works, we give a sufficient condition to guarantee that outliers are stable and provide examples where their fluctuations vary with the particular distribution of the entries of

*Y*or the Jordan decomposition of

*A*. We also exhibit concrete examples where the outlier eigenvalues converge in distribution to the zeros of a Gaussian analytic function.

**Gaetan Borot,**

**Loop equations and all-order asymptotics in mean field models with Coulomb repulsion**

**.**

Abstract.As a generalization of the classical random matrix ensembles, we consider the statistical mechanics of

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*N*particles on the real line, at temperature 1/β subject to analytic

*k*-body interactions on top of a pairwise 2d Coulomb repulsion, I will describe results obtained away from criticality in a recent work with Guionnet and Kozlowski:

(1) analogue of the central limit theorem for fluctuations of linear statistics

(2) all-order asymptotic expansion of cumulants of linear statistics

(3) all-order asymptotic expansion of the partition function

These results are derived from concentration of measures and the analysis of loop equations. Without going into details of the proof, I will review the idea behind loop equations and highlight a few of their properties that are responsible for the nature of the asymptotics we observe.

**Paul Bourgade, Local quantum unique ergodicity for random matrices.**

Abstract.For generalized Wigner matrices, I will explain a probabilistic version of quantum unique ergodicity at any scale, and gaussianity of the eigenvectors entries. The proof relies on analyzing the effect of the Dyson Brownian motion on eigenstates. Relaxation to equilibrium of the eigenvectors is related to a new multi-particle random walk in a random environment, the eigenvector moment flow. This is joint work with H.-T. Yau.

Abstract.

**Yoann Dabrowski,**

**Free transport for convex potentials**

**.**

Abstract.Recently, Dimitri Shlyakhtenko and Alice Guionnet proved by solving a free Monge Ampere equation that self-adjoint variables having conjugate variables coming from a potential close to quadratic, generate the same

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*C**-algebra as free semicircular variables. In this talk based on a joint work with them, I will explain how one can extend this result for (sufficiently regular) convex potentials. This is based on solving a linearized version of free Monge-Ampere equation and uses resolvent estimates for some free partial differential operators coming from generators of Markovian free SDEs. One of the key technical tools is the use of Haagerup tensor products to estimate iterated free difference quotients, which also enables a unified treatment of some cases of conjugate variables relative to a subalgebra.

**Ken Dykema,**

**Quantum symmetric states**

**.**

Abstract.The quantum symmetric states on the universal free product of a

Abstract.

*C**-algebra

*A*with itself infinitely many times are the states that are invariant under the canonical co-actions of S. Wang’s quantum permutation groups. By a variant of Koestler and Speicher’s noncommutative de Finetti theorem, these are characterized in terms of freeness over the tail algebra. We study some natural sets of quantum symmetric states, and show that they are Choquet simplexes. (Joint work with Klaus Koestler and John Williams, as well as with Yoann Dabrowski and Kunal Mukherjee)

**Roland Friedrich, On the geometry related to Free Probability.**

Abstract.In this talk we shall discuss the geometry of the pro-Lie groups describing

Abstract.

multivariable distributions in free probability. In particular we shall mention the role of

hypo-ellipticity and formulate a conjecture.

**Vadim Gorin,**

**Quantized free convolution via representations of classical Lie groups**

**.**

Abstract.My talk is about the asymptotic representation theory of classical Lie groups as their dimension grows. Recall that one of the definitions of the free convolution links it to the description of the spectrum for the sum of two large independent random hermitian matrices. The representation-theoretic analogue of adding matrices is computing tensor products of representations. I will explain how the latter operation leads to a deformation (quantization) of the free convolution. I will also demonstrate the connection between the quantized free convolution and the conjectural asymptotic freeness of the matrices build out of matrix units acting in different irreducible representations of classical Lie groups.

Abstract.

**F. Alberto Grunbaum,**

**Recurrence properties of quantum walks**

**.**

Abstract.Orthogonal Laurent polynomials on the unit circle and their matrix valued versions have started playing a useful role in the study of quantum walks, a quantum analog of classical random walks. In particular the Schur function associated to a measure on the circle is a natural tool to study recurrence properties of these walks. I will discuss similarities and differences between classical and quantum behaviour.

Abstract.

This is joint work with L. Velazquez, R. Werner, A. Werner, J. Bourgain and J.Wilkening

**Alice Guionnet, Approximate transport maps and universality in Random matrix theory.**

**Abstract.**Joint work with F. Bekerman and A. Figalli.

**Mike Hartglass, C*-algebras from weighted graphs.**

Abstract.In this talk, I will sketch the construction, initially appearing in work of Guionnet, Jones, and Shlyakhtenko, of a canonical finite

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*C**-algebra associated to a weighted undirected graph. I will show many properties of this

*C**-algebra including its

*KK*-groups, conditions on when it is simple with unique trace, and conditions on when it embeds into a Cuntz-Krieger graph

*C**-algebra. At the end of the talk, I will sketch some applications of this algebra to graded algebras from planar algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

**Benjamin Hayes, Fuglede-Kadison Determinants and Sofic Entropy.**

**Abstract.**We discuss a general relationship between Fuglede-Kadison and topological entropy of algebraic actions (i.e. an action on a compact metrizable abelian group by automorphisms) for sofic groups. Our results generalize those of Hanfeng Li and Andreas Thom from the amenable groups to sofic groups. We will discuss the new techniques present in the proof, which are the first to avoid a nontrivial determinant approximation. Time alloted, we will discuss some consequences including the following result: if

*G*is a countable discrete sofic group, and

*f*in

*M*

_{n}(

**Z**(

*G*)) is injective on ℓ

^{2}(

*G*)

^{n}, and the inverse of

*f*(as a measurable operator) is in

*M*

_{n}(ℓ

^{2}(

*G*)), but not in

*M*

_{n}(

**Z**(

*G*)), then det

_{L(G)}(

*f*)>1.

**Vaughan Jones, Do all subfactors come from conformal field theory?**

Abstract.Subfactors of finite depth have recently been classifed up to index 5 and a bit beyond.

Abstract.

Most of them are relatively easy to describe but there are few standouts. Evans and Gannon

have found evidene that two of the standouts have some kind of modular behaviour suggesting

the proximity of conformal field theory-though not one that is already known. We will describe

a naive attempt to take a contiuum limit of the planar algebra structure coming from a subfactor

as a first shot at a conformal field theory. The Thompson groups will appear.

**Todd Kemp,**

**Fluctuations of Brownian Motion on**

*GL*_{N}

**.**

Abstract.The Brownian motion on the (complex) Lie group

Abstract.

*GL*

_{N}converges, as a process, in noncommutative distribution as

*N*

*→∞*to a free multiplicative Brownian motion. This was conjectured by Biane in 1997; I recently proved it. I will outline this result, and then discuss the fluctuations (also known as second-order distribution): the rate of convergence is

*O*(1/

*N*), and any collection of moments rescaled appropriately converge to a Gaussian process with covariance that can be described in terms of certain mixed-moments of three freely independent free multiplicative Brownian motions. This is joint work with Guillaume Cebron.

**Claus Koestler, From distributional symmetries to central limit laws.**

Abstract.Recently we have transferred de Finetti type results from classical to noncommutative probability. I will show that the related distributional symmetries are strong enough to provide central limit theorems in an operator algebraic setting. Furthermore I will give some examples and finally address some open problems.

Abstract.

**Franz Lehner, Cumulants for Spreadibility Systems.**

Abstract.Spreadability systems generalize exchangeability systems in the same way quasisymmetric functions generalize symmetric functions. We provide a combinatorial framework and "cumulants" for this setting,

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in particular including monotone cumulants. As an application, we give yet another derivation of the Baker-Campbell-Hausdorff series. This is joint work with T. Hasebe.

**Romuald Lenczewski,**

**Matricial**

*R***-circular systems of operators and random matrices**

**.**

**Abstract.**I will discuss certain systems of operators on the matricially free Fock space which describe the asymptotic ✻-distributions of blocks of large random matrices with independent block-identically distributed (i.b.i.d.) Gaussian entries under the expectation of partial traces when the sizes of these matrices tend to infinity. This discussion will focus on matricial

*R*-circular systems of operators which play the role of matricial analogs of circular operators. They are obtained from canonical decompositions of matricial circular systems describing the asymptotic joint ✻-distributions of symmetric blocks in a similar framework.

**Adam Merberg,**

**Generalized Brownian motion with multiple processes**

**.**

Abstract.We define a notion of generalized Brownian motion with multiple processes similar to that of Guta. We then consider the generalized Brownian motions connected to certain pairs of representations of the infinite symmetric group. We also generalize Guta's

Abstract.

*q*-product of generalized Brownian motion to a product corresponding to a matrix

*q*

_{ij}and show that this product satisfies a central limit theorem.

**James Mingo,**

**Freeness and the Transpose**

**.**

Abstract.Voiculescu showed that independent unitarily invariant random matrices are asymptotically free. Recently Mihai Popa and I showed that unitarily invariant matrix is asymptotically free from its transpose. In this talk I will extend this result to the case of partial transposes.

Abstract.

**Brent Nelson,**

**Free monotone transport without a trace**

**.**

Abstract. In their paper, "Free monotone transport," Guionnet and Shlyakhtenko solve a free analogue of the Monge-Ampère equation to produce a non-commutative version of Brenier's monotone transport theorem. One application of this result is that for sufficiently small |

Abstract

*q*| the

*q*-deformed free group factor Γ

_{q}(

**R**

^{n}) is isomorphic to the free group factor Γ(

**R**

^{n}). By developing this theory in the non-tracial setting we have shown that given a strongly continuous one-parameter group of orthogonal transformations

*U*

_{t}on

**R**

^{n}and sufficiently small |

*q*| the

*q*-deformed Araki-Woods algebra Γ

_{q}(

**R**

^{n},

*U*

_{t}) is isomorphic to the free Araki-Woods factor Γ(

**R**

^{n},

*U*

_{t}). Furthermore, the fixed point argument used to prove this isomorphism can be adapted to the planar algebra setting and consequently allows us to solve the Schwinger-Dyson equation on a graded algebra

*Gr*

_{0}(

*P*). In this talk we will give an overview of these results and outline the free probability methods used to obtain them.

**Alexandru Nica, Double-ended queues and joint moments of left-right canonical operators on full Fock space.**

Abstract.I will present a recent joint work with Mitja Mastnak (arXiv:1312.0269) where we study mixed moments of a (2

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*d*)-tuple of canonical operators on the full Fock space over a

*d*-dimensional space, with d of the operators acting on the left and the other

*d*

acting on the right. The joint action of the (2

*d*)-tuple on the vacuum vector can be described by using the concept of a double-ended

queue. Based on the formula found for the mixed moments of the above (2

*d*)-tuple, we propose a concept of "left-right cumulant functionals" which can be defined on any noncommutative probability space, and which may be helpful for studying the concept of bi-freeness introduced in recent work of Voiculescu.

**Ivan Nourdin, Limit theorems for multiple integrals: an overview of recent results and some open problems.**

Abstract.Starting from a seminal paper of Nualart and Peccati (Ann. Probab., 2005), in recent years many efforts have been made in order to characterize limit theorems for random variables having the specific form of a multiple integral. In this talk, I will review some of the contributions obtained for non-commutative random variables. And if time permits, I will conclude my presentation by providing a list of open problems.

Abstract.

Jonathan Novak,

Jonathan Novak,

**Remarks on the Edrei-Voiculescu theorem**

**.**

Abstract.The Edrei-Voiculesu theorem gives an explicit description of the extreme characters of

Abstract.

*U*(

*∞*). There are two different approaches to its proof: reduction to the classification of two-way totally positive sequences (Voiculescu); approximation by extreme characters of

*U*(

*N*) (Vershik-Kerov). In the second approach, one uses the Weyl character formula and this becomes the asymptotic analysis of Schur polynomials. I will describe an alternative sub-approach which instead uses the Kirillov character formula, and leads to the asymptotic analysis of orbital integrals. I will illustrate this method in the simpler setting of the classification of ergodic invariant measures on the space of infinite Hermitian matrices.

**David Renfrew,**

**Products of random matrices**

**.**

Abstract.In "On the Asymptotic Spectrum of Products of Independent Random Matrices" Götze and Tikhomirov prove that the eigenvalues of product of

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*k*independent random matrices with i.i.d. entries converge to the

*k*-th power of the circular law. We show this is the limiting measure for products of a wider class of random matrices. Joint work with S. O'Rourke, A. Soshnikov and V. Vu.

**Dima Shlyakhtenko, Large**

*N***limits of Random Matrix Models.**

Abstract.In the case of a single matrix, the characterizations of the limit distribution of a random matrix model is well-understood. However, the question for several matrices is quite hard, in the absence of any convexity assumptions on the potential of the model. We give an example showing that limit points of a random multi-matrix model may fail to satisfy the multi-variable analog of the Schwinger-Dyson equation.

Abstract.

Paul Skoufranis,

Paul Skoufranis,

**On Two-Faced Families of Non-Commutative Random Variables**

**.**

Abstract.Recently, Voiculescu introduced the notion of a bi-free two-faced family of non-commutative random variables. In an alternate approach to this theory, Mastnak and Nica defined the (

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*l*,

*r*)-cumulants and defined a two-faced family to be combinatorially-bi-free if the mixed (

*l*,

*r*)-cumulants vanished. In this talk we will demonstrate that the notions of bi-free independence and combinatorial-bi-free independence are equivalent using a diagrammatic view of `bi-non-crossing partitions'. In addition, these partitions produce an operator model on a Fock space for any two-faced family and the expected formulas for the multiplicative convolution of bi-free two-faced families. This is joint work with Ian Charlesworth and Brent Nelson.

**Roland Speicher,**

**Polynomials in free variables**

**.**

Abstract.I will describe the ideas which allow to use operator-valued free probability to calculate the distribution of polynomials

Abstract.

in free variables, hence also the asymptotic eigenvalue distribution of polynomials in asymptotically free random matrices. Main tools are the linearization trick in the form presented by Greg Anderson, recent joint results with Belinschi and Mai on subordination description of operator-valued free convolutions, and work in progress with Belinschi and Sniady on a rigorous version of the hermitization trick to deal with the Brown measure of non-selfadjoint polynomials.

**Yoshimichi Ueda,**

**Orbital free entropy and its Legendre transform approach**

**.**

Abstract.After brief review on previous studies of orbital free entropy, I’ll explain its Legendre transform approach and related questions. The approach fits the previous studies of large

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*N*limits of unitary matrix integrals due to Collins, Guionnet and Segala. (This talk is based on a joint work with F. Hiai.)

**Dan Voiculescu,**

**Bi-free convolution in the plane and the simplest bi-free**

*R***-transform**

**.**

Abstract.I recently extended free probability to systems with left and right variables. For the addition of bi-free pairs of tuples of variables, corresponding bi-free cumulants exist as a consequence of general Lie-theory considerations. For a pair of singletons, there is an explicit formula for the 2-variables generating series of the simplest such cumulants. In particular, this provides the analytic machinery for the computation of bi-free convolutions of probability measures in the plane.

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**John Williams, Non-Commutative Functions and Nevanlinna's representation.**

Abstract.In this talk, we classify the

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*F*-transforms associated to

*B*-valued distributions and the Voiculescu transforms of infinitely divisible

*B*-valued distributions in terms of their analytic and asymptotic properties. This allows for a function theoretic approach to operator valued free probability theory. Time-permitting, we will consider generalizations of Nevanlinna's theorem classifying self maps of upper half space to more general self maps of non-commutative upper half planes.

**Lauren Williams, Positroid and non-crossing partitions.**

Abstract.We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be uniquely constructed by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. And we prove that the probability that a positroid on [

Abstract.

*n*] is connected equals 1/

*e*

^{2}asymptotically. This is joint work with Federico Ardila and Felipe Rincon.

**Makoto Yamashita,**

**Poisson boundary of monoidal categories**

**.**

Abstract.Izumi initiated the study of Poisson boundary for discrete quantum groups in order to analyze the quantum group symmetry on von Neumann algebras. We formulate this notion in the context of semisimple

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*C**-tensor category with conjugates, and obtain a characterization of amenability in terms of the boundary triviality. This setup clarifies the relationship of the Poisson boundaries with several constructions in subfactor theory such as Popa’s standard model and the Longo-Roberts construction. As an application, we obtain a classification of 2-cocycles on the discrete dual of coamenable quantum groups. This talk is based on work in progress with Sergey Neshveyev.

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