Research seminar Free Probability Theory
In this research seminar we treat topics ranging from free probability and random matrix theory to combinatorics, operator algebras, functional analysis and quantum groups. |
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Talks in 2011
- 19.10.2011 Victor Perez Abreu
On an infinitely divisible Gamma random matrix
A new example of infinitely divisible positive definite Gamma random matrix is introduced.
It has properties which make it appealing for modeling under an infinite divisibility framework.
Relations with the Wishart random matrix are pointed out. Finally we will present some results
on the asymptotic spectral distribution of this Gamma random matrix. This is joint work in
progress with Robert Stelzer.
- 26.10.2011 Octavio Arizmendi Echegaray
Free subordinators and the square of a symmetric free infinitely divisible distributions
In this talk we will treat the class of freely infinitely divisible distributions corresponding to
the classical infinitely divisible distributions with positive support: the free regular infinitely
divisible measures. We clarify many descriptions of this class, including their role in free levy
processes: they correspond to free subordinators. We will show that the square of a symmetric
free infinitely divisible distribution is also freely infinitely divisible. Moreover it can be represented
as the free multiplicative convolution of a free poisson and a free regular measure. This gives two
new explicit examples of measures which are infinitely divisible with respect to both classical and
free convolutions: $ \chi^2$ and $ F(1,1)$.
We also prove that the class of free regular measures is closed under free multiplicative convolution.
Furthermore, we show that other operations in non-commutative probability preserve free regular
infinite divisibility such as $t$th Boolean power for $0\leq t\leq 1$, the $t$th free multiplicative
power for $t\geq 1 $ and weak convergence.
Finally, we give some examples and conjectures regarding free regular infinite divisibility.
- 30.11.2011 Michael Stolz
Stein's method and the multivariate CLT for traces of powers on the classical compact groups
Let $M_n$ be a random element of the unitary, special orthogonal, or unitary symplectic groups,
distributed according to Haar measure. By a classical result of Diaconis and Shahshahani, for large
matrix size $n$, the vector $Tr(M_n), Tr(M_n^2), \ldots, Tr(M_n^d))$ tends to a vector of independent
(real or complex) Gaussian random variables. Recently, Jason Fulman has demonstrated that for a single
power $j$ (which may grow with $n$), a speed of convergence result may be obtained via Stein's method
of exchangeable pairs.
In this talk, I will give a leisurely introduction to Stein's method for normal approximation and explain
a multivariate version of Fulman's result (joint work with Christian Döbler, to appear in EJP).
Aktualisiert am: 6. November 2017 Moritz Weber