and Random Matrices

Winter 2020 (it's been a long one)

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Thursday, **July 16**, 10:30 - 11:30, via Zoom

Ian Charlesworth (UC Berkeley)

The free Stein dimension

Regularity questions in free probability ask what can be
learned about a tracial von Neumann algebra from
probabilistic-flavoured qualities of a set of
generators. Non-microstates free entropy theory was
introduced by Voiculescu in the 1990's, and attempts to
study properties arising from derivations defined on
such a set of generators valued in the tensor product of
the $L^2$ space they generate. The free Stein dimension
is a related quantity, which measures the ease with
which such derivations can be found. I will briefly
introduce this quantity and give a summary of some of
our results, before focusing on the details of our proof
of algebraic independence and how the space of
derivations plays a role.
This is joint work with Brent Nelson.

Friday, **June 12**, 11:00 - 12:00, via Zoom

Paul Skoufranis (York)

Bi-Random Matrices and Microstate Free Entropy

Random matrices and free probability have a very
intimate connection that has led to many advances in
both areas. In particular, ideas from random matrices
formed the basis of Voiculescu's microstate free entropy
which has had a profound impact on free probability and
operator algebras. In this talk, we will discuss the
analogues of these ideas in the bi-free setting. An
in-depth analysis of how random matrices behave in the
bi-free setting will be discussed and this leads into
joint work with Ian Charlesworth on microstate bi-free
entropy. Many properties of microstate bi-free entropy
are discussed, including entropy dimension.
Furthermore, using an orbital version of bi-free
entropy, a characterization of bi-freeness is obtained.

Friday, **May 1**, 10:30 - 12:00, via Zoom

Kamil Szpojankowski (Warszawa)

Entry Permutations of Haar Unitary random matrices

In this talk I shall discuss in detail some threads
introduced in the previous talk by M. Popa. In
particular I will focus on entry permutations of Haar
Unitary random matrices. The first part of the talk
will be devoted to a discussion of known results
about spectral behaviour of permuted random
matrices. Next we will switch to the study of
asymptotic $*$--distribution of permuted Haar Unitary
random matrices. We will present sufficient
conditions for a sequence of permutations under which
the sequence of permuted Haar Unitary matrices is
asymptotically circular and free from the not
permuted matrix. It turns out that the conditions for
a sequence of permutations mentioned above are
generic, in the sense that are almost surely
satisfied by a sequence of random, uniformly chosen,
permutations. If time permits I will present ideas
behind the proofs. The talk is based on joint work
with M. Popa.

Friday, **April 24**, 10:30 - 12:00, via Zoom

Mihai Popa (San Antonio)

Entry permutations, asymptotic distributions and asymptotic free independence for several classes of random matrices

Some years ago, together with J. A. Mingo, we showed
that ensembles of unitarilyy invariant random matrices
are asymptotically free from their transposes. The
question "How special is the transpose?" arises then
naturally. That is, given an ensemble $( A_N)_N$ of
random matrices and a sequence of entry permutations
$( \sigma_N )_N$, can we formulate conditions such that
the initial ensemble $( A_N)_N$ and the one consisting
on matrices with permuted entries $(A_N^{ \sigma_N
})_N$, are asymptotically free? Also, what are the
possible limit distributions for the matrices with
permuted entries? The lecture will present some
recent progresses in this problem as well as many
still open questions. Many of the results that will
be presented are joint work with K. Szpojankowski and
J.A. Mingo.

Friday, **April 17**, 10:30 - 12:00, via Zoom

Jamie Mingo (Queen's)

Diagonalizing the Fluctuations of the Gaussian Unitary Ensemble (GUE)

It is an old result of Johansson (1998) that the
fluctuations of the GUE, $X_N$, are diagonalized by
the Chebyshev polynomials $\{C_n\}_{n \geq 0}$. This means
that the covariances of the random variables
$\{\mathrm{Tr}(C_n(X_N^n))\}_n$ are asymptotically
(as $N \rightarrow \infty$) Gaussian and independent. I
will give a combinatorial proof using planar
diagrams. (Notes)

Tuesday, **March 17**, 1:30 - 2:30, Jeff 312

Pei-Lun Tseng (Queen's)

Operator-valued infinitesimal convolutions

In this talk, we will introduce the operator-valued
setting of infinitesimal Boolean and infinitesimal
monotone, and derive the corresponding convolutions.

Tuesday, **March 10**, 1:30 - 2:30, Jeff 422

Pei-Lun Tseng (Queen's)

Infinitesimal Central Limit Theorem

In this talk, we will review the four notions of
infinitesimal independence, and derive associated
central limit theorems.

Tuesday, **March 3**, 1:30 - 2:30, Jeff 422

Daniel Perales Anaya (Waterloo)

Relations between infinitesimal non-commutative cumulants

Boolean, free and monotone cumulants as well as
relations among them, have proven to be important in
the study of non-commutative probability theory (e.g,
using Boolean cumulants to study free infinite
divisibility via the Boolean Bercovici-Pata
bijection). On the other hand, we have the concept of
infinitesimally free cumulants (and the analogue
concept for Boolean and monotone under the name of
cumulants for differential independence).

In this talk we first are going to discuss the basic properties of non-commutative infinitesimal cumulants. Then we will use the Grassmann algebra to show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Later we will review how the relations between the various types of cumulants are captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. Finally, we will observe how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space, and we will present some interesting formulas in this setting. This is a joint work with Adrian Celestino and Kurusch Ebrahimi-Fard.

Tuesday, **February 25**, 1:30 - 2:30, Jeff 422

Jamie Mingo (Queen’s)

Free Compression of Bernoulli Random Variables

If we represent a Bernoulli random variable by a
diagonal matrix with $\pm 1$ entries (half $1$, half
$-1$) and then randomly rotate it with an orthogonal
matrix, we can then cut out the matrix in the upper
left hand corner, of arbitrary size. This is the free
compression of an Bernoulli random variable
(approximately). This compression has the distribution
of the sum of several free Bernoulli random variables,
including fractionally many. We will relate this to the
Kesten-McKay law for random regular graphs.

Tuesday, **February 11**, 1:00 - 2:00, Jeff 202

Jamie Mingo (Queen’s)

A Survey of the Weingarten Calculus

The Weingarten calculus is a method for calculating the
expectation of products of entries of Haar distributed
unitary (or orthogonal) matrices. The basic tool is
Schur-Weyl duality, however I will present an approach
based on Jucys-Murphy operators. The Weingarten
calculus started with Don Weingarten in 1978. Many
authors have contributed since then. This talk will be
based on papers of Collins, Sniady, and Zinn-Justin.

Tuesday, **February 4**, 1:00 - 2:00, Jeff 202

Iris Stephanie Arenas Longoria (Queen’s)

The Moments of the Kesten-McKay Law, II

This will continue from last week.

Tuesday, **January 28**, 12:30 - 1:30, Jeff 202

Iris Stephanie Arenas Longoria (Queen’s)

The Moments of the Kesten-McKay Law

I will present a moment formula for the Kesten-McKay Law for random regular graphs.

Monday, **January 20**, 4:00 - 5:20, Jeff 202

Jamie Mingo (Queen's)

The Distribution of the Partial Transpose of a Haar Unitary Matrix

Mihai Popa and I showed a while ago how the partial
transpose of a matrix can be asymptotically free from
the original matrix. In this talk I will look at the
Banica-Nechita regime (where the number of blocks is
fixed but the size of a block grows) of a Haar
distributed random unitary matrix and compute its free
cumulants. The main tools are Schur-Weyl duality, some
basic facts about the symmetric group and non-crossing
partitions. This is joint work with Mihai Popa and
Kamil Szpojankowski.

Wednesday, **January 8**, 4:30 - 6:00, Jeff 422

Ian Charlesworth (U.C. Berkeley)

Combinatorics of the bi-free Segal-Bargmann transform

The Segal-Bargmann transform provides an isomorphism
between the $L^2$ space of a real Gaussian random
variable and the holomorphic $L^2$ space of a complex
one. It was adapted to the free setting by Biane; an
adaptation to the setting of bi-free probability is the
subject of an ongoing project of mine with Ching Wei Ho
and Todd Kemp. There are a number of peculiarities in
this setting arising from the fact that a central limit
object in bi-free probability is a pair of variables,
specified by two variances and a covariance, leading to
a family of transforms with more parameters. In this
talk I will give an overview of our results,
highlighting in particular a combinatorial argument
based in context free grammars as a tool for
enumeration.

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