Seminar on Free Probability
and Random Matrices

Winter 2018

In the last two decades, many formulas for the asymptotic covariance of linear statistics of random matrices have been found. In the case of a single random matrix ensemble, the two most popular kind of formulas use either difference-quotients or derivatives. During the past year, in joint work with Jamie Mingo, we obtained evidence regarding the generality of the latter option. In this talk we will review recent evidence coming from matricial Gaussian processes further supporting the generality of derivative formulas. This is joint work with Arturo Jaramillo, Juan-Carlos Pardo, and Jose-Luis Perez.

For a tracial $*$-probability space $(\mathcal{A}, \varphi)$, Cébron, Dahlqvist, and Male constructed an enveloping traffic space $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ that extends the trace $\varphi$. The CDM construction provides a universal object that allows one to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure.

We show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a duality between classical independence and free independence.

We apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of Bryc, Dembo, and Jiang and of Mingo and Popa. This is joint work with Camille Male.

The properties of the limiting non-commutative distribution of random matrices can be usually understood thanks to the symmetry of the model, e.g. Voiculescu's asymptotic free independence occurs for random matrices invariant in law by conjugation by unitary matrices. The study of random matrices invariant in law by conjugation by permutation matrices requires an extension of free probability, which motivated the speaker to introduce in 2011 the theory of traffics. A traffic is a non-commutative random variable in a space with more structure than a general non-commutative probability space, so that the notion of traffic distribution is richer than the notion of non-commutative distribution. It comes with a notion of independence which is able to encode the different notions of non-commutative independence.

The purpose of this task is to present the motivation of this theory and to play with the notion of traffic independence.

Last week, we introduced how to find a linearization for a given selfadjoint polynomial and showed some properties of this linearization. We will continue our discussion this week and introduce the operator-valued Cauchy transform. Then, we will show the algorithm for finding the distribution of $P$ where $P$ is a selfadjoint polynomial with selfadjoint variables $X$ and $Y$. Based on this method, we will discuss how to extend this algorithm for finding infinitesimal distribution for $P$.

For given infinitesimal distribution of selfadjoint elements $X$, $Y$, and given a selfadjoint polynomial $P$ with variable $X$ and $Y$. The natural question is whether we can write down the precise formula for the infinitesimal distribution of $P$? In 2009 Belinschi and Shlyakhtenko gave a precise formula to solve for the infinitesimal distribution of $P$ for $P(X,Y)=X+Y$. In the talk, we will discuss how to find the formula for an arbitrary polynomial by using the linearization trick.

Since the 1980's, various classes of random matrices with independent entries were used to approximate free independent random variables. But asymptotic freeness of random matrices can occur without independence of entries: in 2012, in a joint work with James Mingo, we showed the (then) surprising result that unitarily invariant random matrices are asymptotically (second order) free from their transpose. And, in a more recent work, we showed that Wishart random matrices are asymptotically free from some of their partial transposes. The lecture will present a development concerning Gaussian random matrices. More precisely, it will describe a rather large class of permutations of entries that induces asymptotic freeness, suggesting that the results mentioned above are particular cases of a more general theory.

I will continue from last week and compute the infinitesimal cumulants.

The Wishart ensemble is the random matrix ensemble used to estimate the covariance matrix of a random vector. Infinitesimal freeness is a generalized independence stronger than freeness but weaker than second order freeness. I will give the infinitesimal distribution of a real Wishart matrix. It is given in terms of planar diagrams which are ‘half’ of a non-crossing annular partition.

If $X_N$ is the $N \times N$ Gaussian Orthogonal Ensemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

If $X_N$ is the $N \times N$ Gaussian Orthogonal Ensemble (GOE) of random matrices, we can expand $\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in $1/N$, often called a genus expansion. Following the celebrated formula of Harer and Zagier for the GUE, Ledoux (2009) found a five term recurrence for the coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show that the coefficient of $1/N$ counts the number of non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.

In the operator-model theory of de Branges and Rovnyak, any completely non-coisometric (CNC) contraction on Hilbert space is represented as the adjoint of the restriction of the backward shift to a de Branges-Rovnyak subspace of the classical (vector-valued) Hardy space of analytic functions in the open unit disk.

We provide a natural extension of this model to the setting of CNC (row) contractions from several copies of a Hilbert space into itself. A canonical extension of Hardy space to several complex dimensions is the Drury-Arveson space, and the appropriate analogue of the adjoint of the restriction of the backward shift to a de Branges-Rovnyak space is a Gleason solution, a row contraction whose adjoint acts as a several-variable difference quotient. Our several-variable model completely characterizes the class of all CNC row contractions which can be represented as (extremal contractive) Gleason solutions for a multi-variable de Branges-Rovnyak subspace of (vector-valued) Drury-Arveson space.

In free probability theory, the role of the semicircle distribution is analogous to that of the normal distribution in classical probability theory. However, the semicircle distribution also shows up in number theory: it governs the distribution of eigenvalues of Hecke operators acting on spaces of modular cusp forms. In this talk, I will give a brief introduction to this theory of Hecke operators and sketch the proof of a result which is a central limit type theorem from classical probability theory, that involves the semicircle measure.