# Another update on the q-Gaussians

There is presently quite some activity around the q-Gaussians, about which I talked in my last post. Tomorrow (i.e., on Monday, April 25) there will be another talk in the UC Berkeley Probabilistic Operator Algebra Seminar on this topic. Mario Klisse from TU Delft will speak on his joint paper On the isomorphism class of q-Gaussian C∗-algebras for infinite variables with Matthijs Borst, Martijn Caspers and Mateusz Wasilewski. Whereas my paper with Akihiro deals only with the finite-dimensional case (and I see not how to extend this to infinite d) they deal with the infinite-dimensional case, and, quite surprisingly, they have a non-isomorphism result: namely that the C*-algebras for q=0 and for other q are not isomorphic. This makes the question for the von Neumann algebras even more interesting. It still could be that the von Neumann algebras are isomorphic, but then by a reason which does not work for the C*-algebras – this would be in contrast to the isomorphism results of Alice and Dima, which show the isomorphism of the von Neumann algebras (for finite d and for small q) by actually showing that the C*-algebras are isomorphic.

I am looking forward to the talk and hope that afterwards I have a better idea what is going on – so stay tuned for further updates.

# A dual and a conjugate system for the q-Gaussians, for all q

Update: On Monday, April 4, I will give an online talk on those results at the UC Berkeley Probabilistic Operator Algebra Seminar.

I have just uploaded the joint paper A dual and conjugate system for q-Gaussians for all q with Akihiro Miyagawa to the arXiv. There we report some new results concerning the q-Gaussian operators and von Neumann algebras. The interesting issue is that we can prove quite a few properties in a uniform way for all q in the open interval -1<q<1.

The canonical commutation and anti-commutation relations are fundamental relations describing bosons and fermions, respectively. In 1991, Marek Bożejko and I considered an interpolation between those bosonic and fermionic relations, depending on a parameter q with $-1\le q \le 1$ (where q=1 corresponds to the bosonic case and q=-1 to the fermionic case): $a_ia_j^*-q a_j^* a_i=\delta_{ij} 1$. These relations can be represented by creation and annihilation operators on a q-deformed Fock space. (Showing that the q-deformed inner product which makes $a_i$ and $a_i^*$ adjoints of each other is indeed an inner product, i.e. positive, was one of the main results in my paper with Marek.) In the paper with Akihiro we consider only the case where the number d of indices is finite.

Since then studying the q-Gaussians $A_i=a_i+a_i^*$ has attracted quite some interest. Especially, the q-Gaussian von Neumann algebras, i.e., the von Neumann algebras generated by the $A_i$, have been studied for many years. One of the basic questions is whether and how those algebras depend on q. The extreme cases q=1 (bosonic) and q=-1 (fermionic) are easy to understand and they are in any case different from the other q in the open interval -1<q<1. The central case q=0 is generated by free semicircular elements and free probability tools give then easily that this case is isomorphic to the free group factor.

So the main question is whether the q-Gaussian algebras are, for -1<q<1, isomorphic to the free group factor. Over the years it has been shown that these algebras share many properties with the free group factors. For instance, for all -1<q<1 the q-Gaussian algebras are II1-factors, non-injective, prime, and have strong solidity. A partial answer to the isomorphism problem was achieved in the breakthrough paper by Guionnet and Shlyakhtenko, who proved that the q-Gaussian algebras are isomorphic to the free group factors for small |q| (where the size of the interval depends on d and goes to zero for $d\to\infty$). However, it is still open whether this is true for all -1<q<1.

In our new paper, we compute a dual system and from this also a conjugate system for q-Gaussians. These notions were introduced by Voiculescu in the context of free entropy and have turned out to carry important information about distributional properties of the considered operators and to have many implications for the generated von Neumann algebras.

Our approach starts from finding a concrete formula for dual systems; those are operators whose commutators with q-Gaussians are exactly the orthogonal projection onto the vacuum vector. If we also normalize such dual operators by requiring that they vanish on the vacuum vector, then the commutator relation gives a recursion, which can be solved in terms of a precise combinatorial formula involving partitions and their number of crossings, where the latter has, however, to be counted in a specific, and different from the usual, way. The main work consists then in showing that the dual operators given in this way have indeed the vacuum vector in the domain of their adjoints. The action of the adjoints of the dual operators on the vacuum gives then, by general results going back to Voiculescu and Shlyakhtenko, the conjugate variables.

One should note that whereas the action of the dual operators on elements in the m-particle space is given by finite sums, going over to the adjoint results, even for their action on the vacuum, necessarily in non-finite sums, i.e., power series expansions. Thus it is crucial to control the convergence of such series in order to get the existence of the conjugate variables. There have been results before on the existence of conjugate variables for the q-Gaussians, by Dabrowski, but those relied on power series expansions which involved coefficients of the form $q^m$ for elements in the m-particle space and thus guaranteed convergence only for small q. In contrast, the precise combinatorial formulas in our work lead to power series expansions which involve coefficients of the form $q^{m(m-1)/2}$. This quadratic form of the exponent is in the end responsible for the fact that our power series expansions converge for all q in the interval (-1,1).

The existence of conjugate systems for all q with -1<q<1 has then, by previous general results, many consequences for all such q (some of them had been known only for the restricted interval of q, some of them for all q, by other methods). We can actually improve on the existence of the conjugate system and show that it satisfies a stronger condition, known as Lipschitz property. This implies then, by general results of Dabrowski, the maximality of the micro-states free entropy dimension of the q-Gaussian operators in the whole interval (-1,1).

Unfortunately, we are not able to use our results for adding anything to the isomorphism problem. However, the fact that the free entropy dimension is maximal for all q in the whole interval is another strong indication that they might all be isomorphic to the free group factor.

# Topological Recursion Meets Free Probability

Before I am getting too lazy and just re-post here information about summer schools or postdoc positions, I should of course also come back to the core of our business, namely to make progress on our main questions and to get excited about it. So there are actually two recent developments about which I am quite excited. Here is the first one, the second will come in the next post.

During the last few years there was an increasing belief that free probability (at least its higher order versions) and the theory of topological recursion should be related, maybe even just different sides of the same coin. So our communities started to have closer contacts, I started a project on this in our transregional collaborative research centre (SFB-TRR) 195, we had summer schools (here in Tübingen in 2018) and workshops (here in Münster in 2021) on possible interactions and finally there was the breakthrough paper Analytic theory of higher order free cumulants by Gaëtan Borot, Séverin Charbonnier, Elba Garcia-Failde, Felix Leid, Sergey Shadrin. This paper achieves, among other things, the solution to two of our big problems or dreams, namely:

• Rewrite the combinatorial moment-cumulant relations into functional relations between the generating powers series; for first order this was done in Voiculescu’s famous formula relating the Cauchy and the R-transform going back to the beginnings of free probability in the 80s; for second order this was one of the main results in my paper with Benoit, Jamie and Piotr from 2007. For higher orders, however, this was wide open – and its amazing solution can now be found in the mentioned paper.
• Is our theory of free probability only the planar (genus 0) sector of a more general theory which takes all genera into account? This is actually the idea of topological recursion, that you should consider all orders and genera and look for relations among them. I have to admit that I was always quite skeptic about defining the notion of freeness for non-planar situations – but it seems that the paper at hand provides a consistent theory for doing so; apparently also putting the notion of infinitesimial freeness into this setting.

Instead of having me mumbling more about all this, you might go right away to the paper and read its Introduction to get some more precise ideas about what this is all about and what actually is proved.

Let me also add that there is another interesting preprint, On the xy Symmetry of Correlators in Topological Recursion via Loop Insertion Operator by Alexander Hock, which also addresses the functional relations between moments and free cumulants in the g=0 case.

# 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.

Mini-courses:

• 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 https://sites.google.com/view/rmmc2022. 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.

# Queen’s Seminar on Free Probability and Random Matrices

One of the weekly highlights during my times at Queens was the joint seminar with Jamie Mingo on free probability and random matrices. This seminar was of course going on after I left Queen’s, and I also installed different versions of that seminar in Saarbrucken, but at least for me it did not feel the same anymore. So I am happy that we decided now to join forces again and revive our joint seminar in online form. For now it is running on Thursdays at 10 am Eastern time, which is 4 pm German time. You can find the program on the seminar page. If you are interested, write to Jamie to be put on the mailing list.

# Mini-Workshop on Topological Recursion and Combinatorics

There will be an ACPMS mini-workshop on Friday, November 5, 15:00-19:30 (Oslo time) organised by Octavio Arizmendi Echegaray (CIMAT, Guanajuato, Mexico) and Kurusch Ebrahimi-Fard (NTNU Trondheim, Norway). This will be on topolocial recursion and combinatorics, with special emphasis also on the relation with various generalizations of free probability theory.

Title: Topological Recursion and Combinatorics

Topological recursion is a method of finding formulas for an infinite sequence of series or n-forms by means of describing them in a recursive way in terms of genus and boundary points of certain topological surfaces. While topological recursion was originally discovered in Random Matrix Theory, and could be traced back to the Harer-Zagier formula, it was until Chekhov, Eynard and Orantina (2007) that is was systematically studied. Since then it has found applications in different areas in mathematics and physics such as enumerative geometry, volumes of moduli spaces, Gromov-Witten invariants, integrable systems, geometric quantization, mirror symmetry, matrix models, knot theory and string theory. This 1/2-day series of seminar talks aims at exploring combinatorial aspects relevant to the theory of topological recursion in Random Matrix Models and to widen the bridge to free probability and its generalizations such as higher order freeness or infinitesimal freeness more transparent.

Date, time and place

• November 5
• 3:00pm – 7:30pm (Oslo time),
• Zoom (for the link write to the organisers)

Speakers:

• Elba Garcia-Failde (Discussant: Reinier Kramer)
• Séverin Charbonnier (Discussant: Octavio Arizmendi)
• James Mingo (Discussant: Daniel Perales)
• Jonathan Novak (Discussant: Danilo Lewański)

Titles, abstracts and schedule:

https://folk.ntnu.no/kurusche/TRFP

# How Large Must the Kernel of Polynomials in Matrices Be?

Update: If you want to hear more about this, there will actually be an online talk by Guillaume and Octavio on our paper, on Monday, September 13, in the UC Berkeley Probabilistic Operator Algebra Seminar.

Assume I have two symmetric matrices X and Y and I tell you the eigenvalues, counted with multiplicity, of each of them. Then I apply a polynomial P (which is also known to you) to those matrices and ask you to guess the size of the kernel of P(X,Y). If your guess is smaller than the actual size, the Queen of Hearts will pay out your guess in gold; otherwise, if your guess is too large, off with your head. Is there any strategy to survive this for sure and to get out as rich as possible? Let’s say, I don’t even tell you the size of the matrices and only give you the relative number of the eigenvalues, like: the first matrix X has 2/3 of its eigenvalues at 0, 1/6 at 1, and 1/6 at 2; the second matrix Y has 3/4 of its eigenvalues at -1, 1/8 at 0 and 1/8 at +1. What is your guess for the size of the kernel of the anti-commutator P(X,Y)=XY+YX? To be on the safe side you can of course always choose zero; this let’s you survive in any case, but it won’t make you rich. Is there a better guess, which still guarantees you keep your head?

My guess is 5/12. If you want to know how I get this and, in particular, how I can be so sure that this is the best safe bet – have a look on my new paper, joint with Octavio Arizmendi, Guillaume Cebron, Sheng Yin, “Universality of free random variables: atoms for non-commutative rational functions“, which we just uploaded to the arXiv.

If you want to think about the problem before having a look at the paper, take X as before, but Y having now 1/2 of its eigenvalues at -1, 3/8 at 0 and 1/8 at +1. Which of the following is your guess for the size of the kernel of the anti-commutator in this case:

• 5/12
• 1/6
• 1/2
• 1/3

# PhD position for a project on “Free Probability Aspects of Neural Networks” at Saarland University

Update: the position has been filled!

I have funding from the German Science Foundation DFG for a PhD position on the interrelations between free probability, random matrices, and neural networks. Below are more details. See also here for the announcement as a pdf.

Project: Neural networks are, roughly speaking, functions of many parameters in a high-dimensional space and the training of such a network consists in finding the parameters such that the function does what it is supposed to do on the “training inputs”, but also generalizing this in a meaningful way to “real test inputs”. Random matrix and free probability theory are mathematical theories which deal with typical behaviours of functions (which have an underlying matrix structure) in high dimensions and in the limit of large matrix size. Thus it is not surprising that those theories should have something to offer for describing and dealing with neural networks. Accordingly, there have been approaches relying on random matrix and/or free probability theory to investigate questions around deep learning. This interaction between free probability and neural networks is hoped to be bi-directional in the long run. However, the project will not address practical purposes of deep learning; instead we want to take the deep learning challenges as new questions around random matrices and free probability and we aim to develop those theories further on a mathematical level.

Prerequisites: Applicants should have an equivalent of a Master’s degree and a background in at least one of the subjects

• free probability
• random matrices
• neural networks

and an interest in learning the remaining ones and, in particular, in working on their interrelations.

Application: Inquiries and applications should be addressed to Roland Speicher. Your application, in German or in English, should arrive before June 20, 2021. It should contain your curriculum vitae and an abstract of your Master’s thesis. Arrange also for at least one recommendation letter to be sent directly to Roland Speicher, preferably by email. State in your application the name of those you asked for such a letter .

Contact:
Prof. Dr. Roland Speicher
Saarland University
Department of Mathematics
Postfach 15 11 50
66041 Saarbrücken
Germany
speicher@math.uni-sb.de
https://www.uni-saarland.de/lehrstuhl/speicher/

# Another blog on “Free Probability” by Teo Banica

Teo Banica got a bit bored by the lockdown and started to write a series of blogs on various topics, close to his heart and his knowledge – one of them is also one free probability. Check it out here. It’s written in Teo’s personal style, which might seem annoying or provocative to some, but in any case it’s interesting …

Update (September 2020): It seems that Teo got also bored or annoyed of his own blog, so the link above does not work any more … but much of the material has actually been moved to lecture notes and videos. In particular, Teo has the goal of trying to reorganize the quantum group basics, via a series of books. Probably the best to stay updated on this is to check his website or his YouTube channel

# The saga ends …

I have now finished my class on random matrices. The last lecture motivated the notion of (asymptotic) freeness from the point of view of looking on independent GUE random matrices. So you might think that there should now be continuations on free probability and alike coming soon. But actually this part of the story was already written and recorded and if you don’t want to spoil the tension you should watch the series not in its historical but in its logical order:

1. Random Matrices (videos, homepage of class)
2. Free Probability Theory (videos, homepage of class)
3. Non-commutative Distributions (and Operator-Valued free Probability Theory) (videos, homepage of class)

More information, in particular the underlying script (sometimes in a handwritten version, sometimes in a more polished texed version), can be found on the corresponding home page of the lecture series.

May freeness be with you …