# 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.

# Postdoc position on “Integrable Probability” with Alexey Bufetov at Leipzig University

Within the Institute of Mathematics of Leipzig University Professor Alexey Bufetov is looking to fill a postdoctoral research position for up to three years, starting in Autumn 2022. The position is supported by ERC Starting Grant 2021 “Integrable Probability”

The research focus is integrable probability in the wide sense. Experience in one (or more) of the following topics might be of help:

– interacting particle systems,

– random matrices

– models of statistical physics,

– asymptotic representation theory,

– algebraic combinatorics,

– random walks on groups.

The position carries no teaching load. The salary level is TV-L 13.
In order to apply please do the following:

1) ( Required) Send a full CV to the address bufetov@math.uni-leipzig.de

Please include the phrase “Application to a postdoctoral position” and your last name into the subject field.

2) (Optional) You might arrange for several (from one to four) recommendation letters to be sent directly to the address  bufetov@math.uni-leipzig.de

All applications made before 25 March will be fully considered. Late applications will be considered if the position is still vacant.

For informal inquiries please contact    bufetov@math.uni-leipzig.de

# Talk by Moritz Weber in the Wales MPPM Zoom Seminar

The next Wales MPPM Zoom Seminar will given by Moritz Weber (Saarland) on Tuesday, 23rd November at 4.30 pm UK time and UTC.

The title and abstract are:

Easy quantum groups and quantum permutations
Within Woronowicz’s framework of compact quantum groups, there are natural quantum analogs of the symmetric group, the orthogonal group and the unitary group, amongst others. They have in common that their representation theory may be expressed in terms of diagrams. This has been systematically formalized by Banica and Speicher in 2009 within the class of so called “easy” quantum groups.We give an introduction to “easy” quantum groups, their diagrammatic representation theory and we mention some links with Deligne’s interpolation categories. Moreover, we highlight the role of quantum permutations within the theory of quantum automorphism groups of graphs. This also links with nonlocal games in quantum information theory, as we will point out.

Further details, including the programme of upcoming talks, are available on the Wales MPPM Zoom Seminar web page (https://davidemrysevans.wordpress.com/wales-mppm-zoom-seminar/), and some previous talks appear on the Wales MPPM YouTube channel.

# 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/

# Announcement of two talks on free probability at the Technion … and of some more talks

update (from Jan 27): the recordings of the talks of Tobias and mine have been uploaded to youtube, here are the direct links:

I will give a colloquium talk at the Math Department of the Technion, Israel on next Monday, January 25 – online, of course. They have the nice option of a pre-colloquium talk, which provides students with some background for the material appearing in the colloquium talk. Tobias agreed to give such a preparation for my talk. So he will give tomorrow (on Thursday, January 21) an introduction to free probability and its relation with random matrices. Surely a great opportunity for everyone to learn (more) about the subject.

My talk will, of course, have such material in the background, but I tried to prepare it in such a way that even without knowing about free probability one should be able to get the main ideas. So, it might help to know what free semi-circulars are, but it is not necessary (and not assumed) for my talk.

Below are the titles and abstracts of our talks; and here is a link to the Technion page with access information:

Colloquium

update: actually, next week seems to be a busy week for talks around free probability; don’t forget that the UC Berkeley Probabilistic Operator Algebra Seminar will be starting again, on Monday, January 25, with a talk of Friedrich Goetze; and then there will also be a talk by Serban Belinschi on “The Christoffel-Darboux kernel in noncommutative probability” at the Probability Seminar at  Warsaw University of Technology, on Tuesday, January 26.

Tobias Mai: What actually is free probability theory? (Thursday, January 21, 2021)

In my talk, I want to answer this question by giving an introduction to the underlying ideas, basic concepts, and fundamental results of free probability theory. In particular, I will highlight the deep connections of this field with random matrix theory.

Roland Speicher: Singularity of matrices in non-commuting variables and free probability (Monday, January 25, 2021)

The Edmonds’ problem asks to decide about the singularity of a given matrix with linear polynomials in commuting variables as entries, or more general to compute the rank of such a matrix over the field of rational functions. This problem has no known deterministic polynomial time algorithm and it relates to fundamental questions in complexity theory.

Recently, there has been much interest in analyzing a non-commutative variant of the Edmonds’ problem, where the entries are linear polynomials in non-commuting variables and the rank is over the field of non-commutative rational functions (aka free skew field). Garg, Gurvits, Oliveira, and Wigderson showed that for this non-commutative Edmonds’ problem there exists a deterministic polynomial time algorithm. This problem has a remarkable number of diverse origins and motivations and I will present in my talk another such manifestation of the problem, arising from the relation with free probability and random matrix theory. In particular, this approach results also in another, quite analytic, algorithm for calculating the non-commutative rank.

This talk is based on joint work with Johannes Hoffmann, Tobias Mai, and Sheng Yin.

# Nothing New on Connes’ Embedding

It’s now almost a year that we have been told that Connes’ embedding conjecture is not a conjecture anymore, but that it’s actually false. In principle, this is great news as it should open totally new playgrounds, with von Neumann algebras never seen before. The only problem is that we still have not seen them. I am sure that many are looking for them but as far as I am aware nobody outside the quantum information community was able to shed more light on the refutation of Connes’ embedding.

As a believer in the power of non-commutative distributions I tried all my arsenal of moments, cumulants, or Cauchy transforms to get a grasp on how such a non-embeddable von Neumann algebra could look like — of course, without any success. But let me say a few more words an some of my thoughts – if only to come up with a bit longer post for the end of the year.

In our non-commutative distribution language, the refutal of Connes’ embedding says that there are operators in a tracial von Neumann algebra whose mixed moments cannot be approximated by moments of matrices with respect to the trace. We have quite a few of distributions in free probability theory, but the main problem in the present context is that all of them usually can be approximated by matrices, and also all available constructions (like taking free products) preserve such approximations (in particular, since we can model free independence via conjugation by unitary random matrices). Very roughly: our constructions of distributions take some input and then produce some distribution — however, if the input is embeddable, then the output will be so, too. Thus I cannot use those constructions directly to make the leap from our known universe to the new ones which should be out there. The only way I see to overcome this obstruction is to look for distributions which create themselves “out of nothing” via such constructions, i.e., for fixed point distributions of those constructions. For such fixed point distributions I see at least no apriori reason to be embeddable.

But is there any way to make this concrete? My naive attempt is to use the transition from moments to cumulants (or, more analyticially, from Cauchy transforms to R-transforms) for this. We know that infinitely divisible distributions (in particular, compound Poisson ones) are given in the form that their free cumulants are essentially the moments of some other distribution. So I am trying to find reasonable fixed points of this mapping, i.e., I am looking for distributions whose cumulants are (up to scaling or shift) the same as their moments. Unfortunately, all concrete such distributions seem to arise via solving the fixed point equation in an iterative way – which is also bad from our embedding point of view, since those iterations also seem to preserve embeddability. So I have to admit complete and utter failure.

Anyhow, if the big dreams are not coming true, one should scale down a bit and see whether anything interesting is left … so let me finally come to something concrete, which might, or might not, have some relevance …

In the case of one variable we are looking on probability measures, and as those can be approximated by discrete measures with uniform weights on the atoms (thus by the distribution of matrices), this situation is not relevant for Connes’ embedding question. However, I wonder whether a fixed point of the moments-to-cumulants mapping in this simple situation has any relevance. The only meaningful mapping in this case seems to be that I take a moment sequence, shift it by 2 and then declare it as a cumulant sequence — necessarily of an infinitely divisible distribution. Working out the fixed point of this mapping gives the following sequence of even moment/cumulants: 1, 1, 3, 14, 84, 596. The Online Encyclopedia of Integer Sequences labels this as A088717 — which gives, though, not much more information than the fixed point equation for the generating power series.

The above moment-cumulant mapping was of course using free cumulants. Doing the same with classical cumulants gives by a not too careful quick calculation the sequence 1, 1, 4, 34, 496, which seems to be https://oeis.org/A002105 — which goes under the name ”reduced tangent numbers”. There are also a couple of links to various papers, which I still have to check …

Okay, I suppose that’s it for now. Any comment on the relevance or meaning of the above numbers, or their probability distributions, would be very welcome – as well, as any news on Connes’ conjecture.