# Seminar on Free Probability and Random Matrices Winter 2020 (it's been a long one)

## Organizer: J. Mingo <!-- // // format date as dd-mmm-yy // example: 12-Jan-99 // function date_ddmmmyy(date) { var d = date.getDate(); var m = date.getMonth() + 1; var y = date.getYear(); // handle different year values // returned by IE and NS in // the year 2000. if(y >= 2000) { y -= 2000; } if(y >= 100) { y -= 100; } // could use splitString() here // but the following method is // more compatible var mmm = ( 1==m)?'Jan':( 2==m)?'Feb':(3==m)?'Mar': ( 4==m)?'Apr':( 5==m)?'May':(6==m)?'Jun': ( 7==m)?'Jul':( 8==m)?'Aug':(9==m)?'Sep': (10==m)?'Oct':(11==m)?'Nov':'Dec'; return "" + (d<10?"0"+d:d) + "-" + mmm + "-" + (y<10?"0"+y:y); } // // get last modified date of the // current document. // function date_lastmodified() { var lmd = document.lastModified; var s = "Unknown"; var d1; // check if we have a valid date // before proceeding if(0 != (d1=Date.parse(lmd))) { s = "" + date_ddmmmyy(new Date(d1)); } return s; } // // finally display the last modified date // as DD-MMM-YY // document.write( "Last modified on " + date_lastmodified() ); // -->

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.

Previous Schedules

Getting to Jeffery Hall from the Hotel Belvedere