Tuesday, **November 26**, 4:30 - 6:00, Jeff 319

Jamie Mingo (Queen's)

The Combinatorics of Subordination, II

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.

Tuesday, **November 19**, 4:30 - 6:00, Jeff 319

Jamie Mingo (Queen's)

The Combinatorics of Subordination

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.

Tuesday, **November 12**, 4:30 - 6:00, Jeff 319

Josué Vázquez (Queen's)

The Pascal Automorphism and its Spectrum, Part III

Tuesday, **November 5**, 4:30 - 6:00, Jeff 319

Josué Vázquez (Queen's)

The Pascal Automorphism and its Spectrum, Part II

Tuesday, **October 29**, 4:30 - 6:00, Jeff 319

Josué Vázquez (Queen's)

The Pascal Automorphism and its Spectrum

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.

Tuesday, **October 22**, 4:30 - 6:00, Jeff 319

Hao-Wei Huang (Queen's)

Supports of measures in free multiplicative convolution
semigroups, III

Tuesday, **October 15**, 4:30 - 6:00, Jeff 319

Hao-Wei Huang (Queen's)

Supports of measures in free multiplicative convolution
semigroups, II

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.

Tuesday, **October 8**, 4:30 - 6:00, Jeff 319

Hao-Wei Huang (Queen's)

Supports of measures in free multiplicative convolution
semigroups

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.

Tuesday, **October 1**, 4:30 - 6:00, Jeff 319

Jamie Mingo, (Queen's)

Asymptotic Freeness and the Transpose, II

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

Tuesday, **September 24**, 4:30 - 6:00, Jeff 319

Jamie Mingo, (Queen's)

Asymptotic Freeness and the Transpose

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.

Tuesday, **September 17**, 4:30 - 6:00, Jeff 319

Mario Diaz, (Queen's)

Angle of Arrival Based Model for Correlated Multiantenna
Wireless Systems and Its Operator-Valued Equivalent

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.

Tuesday, **September 10**, 4:30 - 6:00, Jeff 319

Camille Male, (CNRS Paris 7)

The fluctuation of linear statistics of
eigenvalues of random matrices

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).

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