Category Archives: Meetings

Free probability, between maths and physics

The fun of free probability is that if you think you have seen everything in the subject suddenly new exciting connections are popping up. This happened for example a few months ago with the preprints

Eigenstate Thermalization Hypothesis and Free Probability, by Silvia Pappalardi, Laura Foini, and Jorge Kurchan


Dynamics of Fluctuations in the Open Quantum SSEP and Free Probability, by Ludwig Hruza and Denis Bernard

According to the authors, the occurrence of free probability in both problems has a similar origin: the coarse-graining at microscopic either spatial or energy scales, and the unitary invariance at these microscopic scales. Thus the use of free probability tools promises to be ubiquitous in chaotic or noisy many-body quantum systems.

I still have to have a closer look on these connections and thus I am very excited that there will be great opportunity for learning more about this (and other connections) and discussing it with the authors at a special day at IHP, Paris on 25 January 2023. This is part of a two-day conference “Inhomogeneous Random Structures”.

Wednesday 25 January: Free probability, between maths and physics.
Moderator: Jorge Kurchan (Paris)

Free probability is a flourishing field in probability theory. It deals with non-commutative random variables where one introduces the concept of «freeness» in analogy to «independence» of commuting random variables. On the mathematical side, it has given new tools and a deeper insight into, amongst others, the field of random matrices. On the physics side, it has recently appeared naturally in the context of quantum chaos, where all its implications have not yet been fully worked out.

Speakers: Denis Bernard (Paris), Jean-Philippe Bouchaud (Paris), Laura Foini (Saclay), Alice Guionnet (Lyon), Frederic Patras (Nice), Marc Potters (Paris), Roland Speicher (Saarbrücken)

Summer school on Free Probability, Random Matrices, and Applications from June 6th to June 10th 2022 at the University of Wyoming

Zhuang Niu and Ping Zhong are organizing a summer school on Free Probability, Random Matrices, and Applications from June 6th to June 10th 2022 at the University of Wyoming. This event is planned to be held in person, but all lectures will have hybrid components for remote participation. 

The summer school will bring together leading experts, young researchers, and students working in free probability, operator algebras, random matrices, and related fields with the aim of fostering new communications and collaborations. Four mini-courses will be given by leading researchers in the field. In addition, there will be some survey talks or research talks. The mini-courses are designed for graduate students and young researchers. However, everyone is very welcome to participate in the meeting.  


  • Hari Bercovici (Indiana), Complex analysis in free probability. 
  • Benoit Collins (Kyoto), Around the operator norm convergence of random matrices in free probability.  
  • Ken Dykema (College Station), On spectral distributions and decompositions of operators in finite von Neumann algebras 
  • Alexandru Nica (Waterloo), Recent developments related to the use of cumulants in free probability 

The conference website and registration form can be found at Please indicate in the registration form if you would like to contribute a research talk.  

Financial supports are available, and priority will be given to graduate students, junior researchers, and other participants without travel grants. The deadline of registration for full consideration of a financial aid is April 30th, 2022.  

Please share this information with anyone who might be interested in this meeting. 

Update on videos of lectures and talks

Videos of talks at Fields Institute

The videos of my three talks in the distinguished lecture series at the Fields Institute are now online and can be found here.

In particular, the first talk, entitled Dan-Virgil Voiculescu: visionary operator algebraist and creator of free probability theory, was a talk for a public audience and gives not only a bit of information on Dan Voiculescu, but also a very high level idea of what free probability is all about. And it also has some movie references …

Videos of lecture series “Non-commutative distributions”

The course on “Non-commutative distributions” has now finished; all 20 lectures are online and can be found on our video platform. My hand-written notes for the classes can be found here.

Focus Program on Applications of Noncommutative Functions; Featuring a Celebration Banquet for Dan Voiculescu’s 70th Birthday

Today started the Focus Program on Applications of Noncommutative Functions at the Fields Institute in Toronto. There will be two workshops: this week on the “Developments and Technical Aspects of Free Noncommutative Functions” and next week on “Applications to Random Matrices and Free Probability of Free Noncommutative Functions”. Both workshops look interesting to me; unfortunately I will miss most of the first one as I will fly only on Wednesday to Toronto.

I will give a series of three talks on the relation between free probability and random matrices. The first talk will be quite general and is also intended for a public academic audience. Its main purpose is to celebrate the 70th birthday of Dan Voiculescu by giving an idea of Dan’s achievements and of free probability theory. The talk and the banquet will be on the very day of Dan’s birthday.

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.


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.


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.


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”


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”

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.