Author Archives: rolandspeicher

Free Activities in Montreal, Week 4 and Conclusion

Week 4

The second workshop “Free Probability: the applied perspective”, in the last week of the program, had its focus on more applied directions. In recent years, it has become apparent that free probability, with its tools for calculating the eigenvalue distribution of random matrices or dealing with outliers, might have quite some impact for questions in applied subjects like: physics, statistics, wireless communications, or machine learning. Many of the talks during the second workshop addressed such issues and showed this surprising arch of free probability from the abstract to the very applied: we saw three vignettes on free probability and statistics; heard about a new theory for sketching in linear regression; and learned about free probability for deep learning.

Conclusion

Apart from the exciting and surprising new connections and directions of free probability, there were also a couple of talks which presented new important results per se, creating a lot of discussions and potentially new collaborations. Just to mention a few, the talks “The free field meets free probability theory” by Mai and “The atoms of the free additive convolution of two operator-valued distributions” by Belinschi gave qualitatively new results about one of the hard problems of the analytic theory of free probability, namely the absence or the possible occurrence of atoms for various functions in non-commuting variables. The talk “Traffic independence and freeness over the diagonal” by Cebron highlighted important progress in the theory of traffic independence, by relating the up to now quite combinatorial theory to analytic tools from operator-valued freeness, and thus opening a quite new direction for this important concept. Another impressive advance, on the notoriously difficult task of calculating Brown measure, was presented in the talk “The Brown measure of free multiplicative Brownian
motion” by Kemp.

I think that the whole program on free probability at the CRM was a big success. In particular, the attendance of many young researchers, often for the whole month or a substantial part of it, and the many lively discussions in the seminar rooms and halls of the Pavillon Aisenstadt, following the talks or signalling ongoing or new collaborations, showed clearly that the subject is still very vibrant and full of new ideas. I am quite optimistic about the future of free probability theory.

Free Activities in Montreal, Week 3

Our one-month program on New Developments in Free Probability and Applications has started. After a one-week workshop on the more theoretical side, there will be now two weeks of introductions and survey talks, as well as seminar talks, before we finish with another workshop on the (potential) applications of free probability.

Here is the schedule of talks for week 3, featuring in particular two of the three Aisenstadt Chair talks of Alice Guionnet. The first of those is intended for a general public, see also the CRM website. The third Aisenstadt Chair talk will be the opening talk of the second workshop.

WEDNESDAY March 20

Pavillon Claire-McNicoll, Université de Montréal, Room Z-220

16:15 Alice Guionnet (Aisenstadt Chair 2019)
“Free probability and Random matrices”
Free probability is the natural framework to consider matrices with size going to infinity. Since this key remark was made by Voiculescu in the nineties, these two fields have enriched each other continuously. We will discuss a few of these fruitful crossovers. This talk will only require a general mathematical background.


THURSDAY March 21

Pavillon André-Aisenstadt, Université de Montréal, Room 5340

09:30 – 11:00 Mireille Capitaine
“Introduction to Outliers for Deformed Wigner Matrices”


14:30 – 15:30 Felix Leid
“Maps, Partitioned Permutations, and Free Probability”

16:00 – 17:00 Maxime Fevrier
“From conditional freeness to infinitesimal freeness”


FRIDAY March 22

Pavillon André-Aisenstadt, Université de Montréal, Room 1360

10:30 Alice Guionnet (Aisenstadt Chair 2019)
“Free probability and random matrices: Conjugate variables and the Dyson-Schwinger equations”
In this talk I will discuss some uses of integration by parts in free probability, random matrices and related topics. In particular I will show how it can be used to study the large dimension asymptotics in random matrices and tilings.



Free Activities in Montreal, Week 2

Our one-month program on New Developments in Free Probability and Applications has started. After a one-week workshop on the more theoretical side, there will be now two weeks of introductions and survey talks, as well as seminar talks, before we finish with another workshop on the (potential) applications of free probability.

Here is the schedule of talks for this week; they will be held in room 5340, 5th floor, Andre-Aisenstadt building on the campus of the Universite de Montreal.

Note that the second talk on Friday has changed; Felix Leid’s talk has been moved to next week, Ben Hayes will talk instead.

TUESDAY March 12

9:30 – 12:30 Tobias Mai, Roland Speicher, and Sheng Yin
“Introduction to Regularity and the Free Field”

Lunch break

2:30 – 4:00 Hari Bercovici
“Introduction to outliers”


THURSDAY March 14

9:30 – 12:30 Benson Au, Guillaume Cébron, and Camille Male
“Introduction to Traffic Freeness”

Lunch break

2:30 – 3:30 Laura Maassen
“Group-theoretical quantum groups”

4:00 – 5:00 Simon Schmidt
“Quantum automorphism groups of finite graphs”


FRIDAY March 15

9:30 – 10:30 Josué Váquez Becerra
“Fluctuation moments induced by some asymptotically liberating unitary matrices”

11:00 – 12:00 Ben Hayes
“A random matrix approach to the Peterson-Thom conjecture”




Update on “Free Probability” Class

The lectures on our free probability class are over now. All 26 lectures are uploaded on the video platform, and the scans of my handwritten course notes can be found here.

I plan to continue next term with a class on “Non-Commutative Distributions”; this will in particular cover the operator-valued version of free probability and its use for dealing with polynomials in free variables, as well as addressing regularity properties of the distribution of such polynomials. There will be more cool stuff, but I still have to think about details. Our summer term starts in April, then I will be back with more information.

The plan is to continue with the recording of the lectures. If you have any suggestions on how to improve on this, please let me know.

Asymptotic Freeness of Wigner Matrices

When preparing my lectures for the asymptotic freeness of various random matrix ensembles I stumbled about the situation concerning Wigner matrices. We all know that Wigner matrices and deterministic matrices are asymptotically free, but all the proofs I am aware of are annoyingly complicated. Shouldn’t there be a nice and simple proof without too many technicalities?

As was said in the section “Open problems and possible future directions” of the report of the 2008 Banff workshop “Free Probability, Extensions, and Applications”: “Whereas engineers have no problems in applying asymptotic freeness results for unitarily invariant ensembles it has become apparent that they do not have the same confidence in the analogous results for Wigner matrices. The main reason for this is the lack of precise statements on this in the literature. This has to be remedied in the future.”

Actually, at that time there existed already some results in this direction in the paper On Certain Free Product Factors via an Extended Matrix Model from 1993 of Ken Dykema. There the asymptotic freeness between independent Wigner matrices and diagonal (or even more general: block diagonal) deterministic matrices had been shown. But the case of general deterministic matrices remained open. Taking on the challenge by the engineers, we were determined to write down nice and accessible proofs, also including the full deterministic case.

The result of this was that such statements and proofs were included in the book An Introduction to Random Matrices of Greg Anderson, Alice Guionnet, and Ofer Zeitouni on one side and in my book with Jamie on the other side. However, I have to admit what looked like an easy exercise to Jamie and me at the beginning turned out to me much more complicated. An intermediate outcome of this was my paper with Jamie on Sharp Bounds for Sums Associated to Graphs of Matrices, but even with this as a nice black box the final proof still required a couple of technical pages in our book.

So I would like to come back to the original challenge and want to see what we really know about the relation between Wigner matrices and deterministic matrices. What are the clean statements and what are nice proofs. The situation for Wigner matrices is also more complicated than for Gaussian matrices, as the real and imaginary part for Wigner matrices do not have to be independent, hence the complex situation cannot be directly reduced to the real one, and questions about the *-freeness of non-selfadjoint Wigner matrices is also not exactly the same as the freeness of selfadjoint ones. Of course, all is related and similar, but if one asks a concrete question, usually it is hard to find the answer exactly for this in the literature.

I hope to collect here information about what is out there in the literature on that problem, with the final goal of coming up with the cleanest statements and the simplest proofs. If you have any information or ideas in this context, please let me know.

News on Lectures on “Free Probability Theory”

The lectures on free probability are back!

I have now started to put scans of my handwritten lecture notes online; they correspond more or less to what I write on the blackboard. As we are now, in the context of random matrix calculations, having a lot of indices hanging around it might in some cases be easier to decipher those from my notes than from the blackboard. Maybe sometimes in the future I will even tex them, but don’t count on this … In any case, most of the material I am presenting in this class is either from my book with Andu or from my book with Jamie, so that there exist already nicely written notes on this.

There is of course much more to say about random matrices. One issue is that in this class I cover only convergence and asymptotic freeness results in the averaged sense. Of course, almost sure versions of those results usually also exist. For more on those and other aspects of random matrices I refer to the random matrix literature (part of which you can find on the homepage of my class “Random Matrices” from last term). There exists also a nice tex-ed version of the lectures notes from my class on random matrices.

 

Free Probability Meetings in 2019

2019 will again be a year with quite a few meetings around free probability.

The main event will be a month-long program on New Developments in Free Probability and Applications at CRM in Montreal in March 2019. There will be two workshops: one, at the beginning of March, on the theory and its extensions and the second, at the end of March, on the applied perspective. In the two weeks in between there will also be quite some activity, in particular, we are aiming at bringing graduate students and postdoctoral fellows quickly to the frontiers of the subject. Furthermore, Alice Guionnet will give the Aisenstadt Chair lecture series between both workshops.

This program is part of the year long celebration of the CRM’s 50th anniversary. It seems very appropriate to have such a meeting on the blossoming of free probability theory, and its promise for the future at the place where the seed was sown. In the spring of 1991 Dan Voiculescu was the holder of the Andre Aisenstadt chair at the CRM in Montreal during the ’91 operator algebra program. At this time, free probability was still in its infancy and only known to a small group of enthusiasts. This was going to change. Voiculescu gave the Aisenstadt Lectures on free probability in Montreal, organizing the material and bringing it with the help of his students Ken Dykema and Alexandru Nica into a publishable form. The resulting book was the first volume in the CRM Monograph Series and was instrumental for making the theory more generally accessible and attracting many, in particular young, researchers to the subject. It is still the most cited literature on free probability. Andu, Dan, and Ken (as well as a couple of other experts) will stay as Simons Scholars-in-Residence for the whole program at CRM

Another month-long program with a substantial free probability component will be the Focus Program on Applications of Noncommutative Functions at the Fields Institute in Toronto, June 10 – July 5, 2019. In particular, one of the workshops of the program, June 17-21, deals with applications of noncommutative functions to random matrices and free probability.

The focus program at the Fields Institute will also include a celebratory banquet on June 14, in honour of the 70th birthday of Dan Voiculescu.