# Seminar on Free Probability and Random Matrices Fall 2014

## Organizers: J. Mingo and S. Belinschi <!-- // // 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() ); // -->

Schedule for Current Term

Wednesday, December 3, 4:30 - 6:00, Jeff 222
Hao-Wei Huang (Queen's)
Infinitely divisible matrices and their applications in free probability theory, IV

Tuesday, November 25, 4:30 - 6:00, Jeff 319
Hao-Wei Huang (Queen's)
Infinitely divisible matrices and their applications in free probability theory, III
We will discuss some theorems which characterize freely infinitely divisible laws in terms of Voiculescu transforms. Also to be discussed are freely infinitely divisible laws are characterized via the free Lévy-Hinčin formula.
Tuesday, November 18, 4:30 - 6:00, Jeff 319
Hao-Wei Huang (Queen's)
Infinitely divisible matrices and their applications in free probability theory, II
We will discuss some theorems which characterize freely infinitely divisible laws in terms of Voiculescu transforms. Also to be discussed are freely infinitely divisible laws are characterized via the free Lévy-Hinčin formula.

Tuesday, November 11, 4:30 - 6:00, Jeff 319
Jamie Mingo (Queen's)
Freeness and the Transpose, II
This will be a continuation from November 3.

Monday, November 3, 5:00 - 6:30, Jeff 102
Jamie Mingo (Queen's)
Freeness and the Transpose
M. Popa and I have shown that a random matrix can be free from its transpose if it is unitarily invariant. We have recently taken this one step further and have shown that for a Wishart matrix one gets an asymptotically free family consisting of the original matrix, two partial transposes (left and right), and the full transpose.

Tuesday, October 21, 4:30 - 6:00, Jeff 319
Hao-Wei Huang (Queen's)
Infinitely divisible matrices and their applications in free probability theory
The theory of positive definite matrices, positive definite functions, and positive linear maps is rich in content. The notion of infinitely divisible matrices, a subclass of positive definite matrices, also has been studied for a long time by R. Horn and R. Bhatia. These subjects are closely related to moment sequences, infinitely divisible laws with respect to classical convolution, and non-commutative analysis. We will present some of these results and discuss the applications in free probability theory.

Tuesday, October 14, 4:30 - 6:00, Jeff 319
Josué Daniel Vázquez Becerra (Queen's)
Asymptotic Liberation in Free Probability, II
I will continue from last week.
Tuesday, October 7, 4:30 - 6:00, Jeff 319
Josué Daniel Vázquez Becerra (Queen's)
Asymptotic Liberation in Free Probability
Let $M$ be a finite non-empty set. Suppose that for each $n \in \mathbb{N}$ we are given a family $\{B_{n,m}\}_{m \in M}$ of $n$-by-$n$ complex matrices such that $\displaystyle{ \sup_{n \in \mathbb{N} } \sup_{m\in M}} \Vert B_{n,m} \Vert < \infty$, and the limit $\displaystyle{ \lim_{n\rightarrow \infty } } \frac{1}{n} \mathbf{tr} [ (B_{n,m})^{k} ]$ exists for all $k\in \mathbb{N}$ and all $m \in M$. Can we modify the sequence $\{ \{B_{n,m}\}_{m \in M}\}_{n\in \mathbb{N}}$ of families of complex matrices such that the resulting modified sequence is asymptotically free? For instance, a well-known result in free probability states that if $\{U_{n,m}\}_{m\in M}$ is a family of independent $n$-by-$n$ Haar-distributed random unitary matrices for each $n \in \mathbb{N}$, then the sequence $\{ \{U_{n,m}^{*}B_{n,m}U_{n,m}\}_{m\in M} \}_{n \in \mathbb{N}}$ is asymptotically free. In this talk, we first present the notion of asymptotically liberating sequences of families of random unitary matrices. Then, we show that such sequences can be used to the same end as the sequence of families of independent Haar-distributed random unitary matrices above, i.e., we show that if $\{\{V_{n,m}\}_{m\in M}\}_{n \in \mathbb{N}}$ is asymptotically liberating, then $\{ \{V_{n,m}^{*}B_{n,m}U_{n,m}\}_{m\in M} \}_{n \in \mathbb{N}}$ is asymptotically free. After, we state and prove a theorem on sufficient conditions for a sequences of families of random unitary matrices in order to be asymptotically liberating. Finally, if time permits we provide an example of an asymptotically liberating sequence derived from uniformly distributed random signed permutation matrices and Hadamard matrices.
Friday, October 3, 4:30 - 6:00, Jeff 110
In this talk we will study certain elements in $L^{\infty-}(\Omega,{\mathcal F},{\mathbb P}) \otimes \textrm{M}_d({\mathcal C})$ where $({\mathcal C},\varphi)$ is a non-commutative probability space and $(\Omega,{\mathcal F},{\mathbb P})$ is a classical probability space. In particular, a numerical algorithm to compute the mean $\textrm{ M}_d({\mathbb C})$-valued Cauchy transform of elements of the form $A\circ {\bf X}$ with $A$ a $d\times d$ selfadjoint random matrix and ${\bf X}$ a $d\times d$ selfadjoint operator-valued matrix with free circular entries will be analyzed. Here $\circ$ denotes the Hadamard or pointwise multiplication. Also a central limit theorem for a more general class of elements in $L^{\infty-}(\Omega,{\mathcal F},{\mathbb P}) \otimes \textrm{M}_d({\mathcal C})$ will be derived.