Seminar on Free Probability
and Random Matrices

Fall 2013

The conclusion from last week. The free additive convolution of random variables was originally described by the R and S transform. An alternative approach using subordination of Cauchy transforms was later found. Following Voiculescu and Biane we give a combinatorial interpretation of the subordination function.

The free additive convolution of random variables was originally described by the R and S transform. An alternative approach using subordination of Cauchy transforms was later found. Following Voiculescu and Biane we give a combinatorial interpretation of the subordination function.

In 1981, A. Vershik introduced the concept of an adic transformation as well as the Pascal automorphism as an example. Since the Pascal automorphism is a non-stationary adic transformation whose construction is simple, A. Vershik also suggested the study of its properties. In this context, one question that remained unsolved for thirty years was the type of spectrum that the Pascal automorphism has as an automorphism of Lebesgue spaces. This problem was studied by K. Petersen, K. Schmidt, E. Janvresse, T. de la Rue, X. Méla, A. Lodkin, I. Manaev and A. Minabutdinov but it was A. Vershik who solved it. In this lecture, we present the Pascal automorphism and give the necessary results to determine the type of spectrum that it has.

It is known that the free convolutions of any probability measure $\mu$ on the real line and the semicircular distribution with mean $0$ and variance $t$ have a non-increasing number of components in the supports as $t$ increases. The same property also holds for free multiplicative convolution and the free convolution semigroup. In this talk, I will show that free multiplicative convolution semigroups generated by certain Borel probability measures on the unit circle and on the positive real line have this property. This is joint work with Ping Zhong.

In free probability, there are many cases that the number of components in a family of probability measures with parameter $t$ is a non-increasing function of $t$. For instance, in 1997 Biane showed that the free convolution of any Borel probability measure on $\mathbb{R}$ and semicircular distribution with variance t has this property. Later, he showed that this non-increasing property also holds for the free multiplicative convolution of an arbitrary probability measure on the unit circle with the free multiplicative analogues of the normal distribution on the unit circle. In fact, the partially defined free additive convolution semigroup generated by any Borel probability measure on $\mathbb{R}$ has this property as well. In this talk, we will talk about our recent paper showing that the partially defined free multiplicative convolution semigroups generated by any Borel probability measure on the unit circle and on the positive real line have similar results. This is joint work with Ping Zhong.

This week I will show that $U$ is asymptotically *-free from $U^t$ where $U$ is a Haar distributed random unitary matrix.

In most asymptotic freeness theorems in random matrix theory one assumes that the entries of the two ensembles are independent plus some extra symmetry condition. I will give two surprising examples where a random matrix is asymptotically free from its transpose. This is joint work with Mihai Popa and Roland Speicher.

In the context of multiantenna wireless systems, one important question is that regarding the system’s scalability, i.e. the system’s capability to increase its capacity as it becomes larger. In this talk we will give a model to calculate the asymptotic capacity, with respect to the number of antennas, of correlated multiantenna systems. This model depends on the angle of arrival of the antennas that compose the system. Surprisingly, this leads to an operator-valued model which can be thought as the operator valued version of the standard Kronecker model. This work was part of my M.Sc. thesis under supervision of Dr. Víctor Pérez-Abreu.

The spectra of different random matrix models can constitute models for interacting particles with different degree of repulsion. The case of symmetric of Hermitian random matrix $A_N$ whose entries are i.i.d. entries with small moments (i.e. such that the law $\mu$ of $\sqrt N A_N(i,j)$ does not depend on $N$ and admits moments of any order) is now well understood. Wigner proved the convergence of the normalized linear statistics for these matrices: \[ \frac 1 N \sum_{i=1}^N f(\lambda_i) \mathop{\longrightarrow}_% { N \rightarrow \infty} \int_{-2}^2 f(x) \frac{\sqrt{4-x^2}}{2\pi}\, dx \] where the $\lambda_i$'s are the eigenvalues of $A_N$ and $f$ is a polynomial or a bounded continuous function. Under the assumption that the fourth moment of $\mu$ exists, the fluctuation around their expectation of these statistics were studied by Johnson, Pastur, and Sinai and Soshnikov. It turns out that for sufficiently smooth functions $f$, \[ Z_N(f) = \sum_{i=1}^N f(\lambda_i) - E[ \sum_{i=1}^N f(\lambda_i)] \] tends to a Gaussian random variable whose covariance depends on the first four moments of $\mu$. The absence of normalization by $\sqrt{N}$ shows that the eigenvalues of $A$ fluctuate very little. In this talk, we present two extensions of this result. First for variations of this model where the measure $\mu$ does not have any second moment or when it depends on $N$, with moments growing with $N$. Secondly for random matrices with dependent entries, as adjacency matrices of random graphs. In both cases, one has to normalized the random variable $Z_N(f)$ to get the convergence to a non trivial Gaussian random variable. (In collaboration with Benaych-Georges, Guionnet, and Péché).