Preprints

Asymptotic infinitesimal freeness with amalgamation for Haar quantum
unitary random matrices
Authors: S. Curran and R. Speicher
Abstract: We consider the limiting distribution of $U_NA_NU_N^*$ and
$B_N$ (and more general expressions), where $A_N$ and $B_N$ are $N
\times N$ matrices with entries in a unital C$^*$algebra $\mathcal B$
which have limiting $\mathcal B$valued distributions as $N \to
\infty$, and $U_N$ is a $N \times N$ Haar distributed quantum unitary
random matrix with entries independent from $\mathcal B$. Under a
boundedness assumption, we show that $U_NA_NU_N^*$ and $B_N$ are
asymptotically free with amalgamation over $\mathcal B$. Moreover,
this also holds in the stronger infinitesimal sense of
BelinschiShlyakhtenko.
We provide an example which demonstrates that this example may fail
for classical Haar unitary random matrices when the algebra $\mathcal
B$ is infinitedimensional.

Free Probability Theory
Author: R. Speicher
(contribution for Handbook on Random Matrix Theory, to be published by
Oxford University Press)
Abstract:
Free probability theory was created by Dan Voiculescu around 1985,
motivated by his efforts to understand special classes of von Neumann
algebras. His discovery in 1991 that also random matrices satisfy
asymptotically the freeness relation transformed the theory
dramatically. Not only did this yield spectacular results about the
structure of operator algebras, but it also brought new concepts and
tools into the realm of random matrix theory. In the following we will
give, mostly from the random matrix point of view, a survey on some of
the basic ideas and results of free probability theory.

The normal distribution is $boxplus$infinitely divisible
Authors: S. Belinschi, M. Bozejko, F. Lehner, R. Speicher
Abstract: We prove that the classical normal distribution is
infinitely divisible with respect to the free additive convolution. We
study the Voiculescu transform first by giving a survey of its
combinatorial implications and then analytically, including a proof of
free infinite divisibility. This is only the third example known to us
at this moment of a nontrivial distribution that is infinitely
divisible with respect to both classical and free convolution, the
others being the Cauchy distribution and the free 1/2stable
distribution.

Sharp Bounds for Sums Associated to Graphs of Matrices
Authors: J. Mingo and R. Speicher
Abstract: We provide a simple algorithm for finding the optimal upper
bound for sums of products of matrix entries of the form
S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4}
... t^m_{j_2m1 j_2m} where some of the summation indices are
constrained to be equal. The upper bound is easily obtained from a
graph G associated to the constraints in the sum.

Stochastic aspects of easy quantum groups
Authors: T. Banica, S. Curran, R. Speicher
(to appear in Prob. Th. Related Fields)
Abstract: We consider several orthogonal quantum groups satisfying the
easiness assumption axiomatized in our previous paper. For each of
them we discuss the computation of the asymptotic law of Tr(u^k) with
respect to the Haar measure, u being the fundamental
representation. For the classical groups O_n, S_n we recover in this
way some wellknown results of Diaconis and Shahshahani.

De Finetti theorems for easy quantum groups
Authors: T. Banica, S. Curran, R. Speicher
Abstract: We study sequences of noncommutative random variables which
are invariant under "quantum transformations" coming from an
orthogonal quantum group satisfying the "easiness" condition
axiomatized in our previous paper. For 10 easy quantum groups, we
obtain de Finetti type theorems characterizing the joint distribution
of any infinite, quantum invariant sequence. In particular, we give a
new and unified proof of the classical results of de Finetti and
Freedman for the easy groups S_n, O_n, which is based on the
combinatorial theory of cumulants. We also recover the free de Finetti
theorem of K\"ostler and Speicher, and the characterization of
operatorvalued free semicircular families due to Curran. We consider
also finite sequences, and prove an approximation result in the spirit
of Diaconis and Freedman.

Classification results for easy quantum groups
Authors: T. Banica, S. Curran, R. Speicher
Abstract: We study the orthogonal quantum groups satisfying the
``easiness'' assumption axiomatized in our previous paper, with the
construction of some new examples, and with some partial
classification results. The conjectural conclusion is that the easy
quantum groups consist of the previously known 14 examples, plus of an
hypothetical multiparameter ``hyperoctahedral series'', related to
the complex reflection groups $H_n^s=\mathbb Z_s\wr S_n$. We discuss
as well the general structure, and the computation of asymptotic laws
of characters, for the new quantum groups that we construct.

The noncommutative cycle lemma
Authors: C. Armstrong, J. Mingo, R. Speicher, J. Wilson
(to appear in J. Comb. Th. A)
Abstract: We present a noncommutative version of the cycle lemma of
Dvoretsky and Motzkin that applies to free groups and use this result
to solve a number of problems involving cyclic reduction in the free
group. We also describe an application to random matrices, in
particular the fluctuations of Kesten's Law.

Resolvents of Rdiagonal operators
Authors: U. Haagerup, T. Kemp, R. Speicher
(to appear in Transactions of AMS)
Abstract:
We consider the resolvent $(\lambdaa)^{1}$ of any $R$diagonal
operator $a$ in a $\mathrm{II}_1$factor. Our main theorem gives a
universal asymptotic formula for the norm of such a resolvent. En
route to its proof, we calculate the $R$transform of the operator
$\lambdac^2$ where $c$ is Voiculescu's circular operator, and give
an asymptotic formula for the negative moments of $\lambdaa^2$ for
any $R$diagonal $a$. We use a mixture of complex analytic and
combinatorial techniques, each giving finer information where the
other can give only coarse detail. In particular, we introduce {\em
partition structure diagrams}, a new combinatorial structure arising
in free probability.

On the rate of convergence and BerryEsseen type theorems for a
multivariate free central limit theorem
Author: Roland Speicher
(not intended for publication in this form)
Abstract:
We address the question of a Berry Esseen type theorem for the speed
of convergence in a
multivariate free central limit theorem. For this, we estimate the
difference between the
operatorvalued Cauchy transforms of the normalized partial sums in an
operatorvalued
free central limit theorem and the Cauchy transform of the limiting
operatorvalued
semicircular element.
Roland Speicher.