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# 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.

# Lecture Notes on Non-Commutative Distributions

Waiting has come to an end … finally the pdf edition of the Lecture Notes on Non-Commutative Distributions has arrived. As a bonus for loyal followers I have added, compared to the actual content of the lecture series, two small sections at the end on what our machinery has to say about Connes embedding problem and the q-Gaussian distribution. Though, don’t expect too much there …

# On the origin of moment-cumulant formulas

When I gave a class on free probability theory a few years ago, I thought it would be a good idea to localize evidence for my usual statement that in the classical context the idea of viewing moment-cumulant formulas in terms of (multiplicative functions on) set partitions, as well as the vanishing of mixed cumulants in independent random variables, goes back to Rota; the main reference on this seemed to be the Foundations of Combinatorial Theory papers, part I and part VI. This is at least what I said in my old papers, like here or here, and what Jonathan Novak, for example, also iterates in his nice Three Lectures on Free Probability. But when I tried to find any mentioning of cumulants in those papers of Rota I could not localize anything. Also in the paper of Rota with Shen, On the Combinatorics of Cumulants, there is no clear mentioning of vanishing of mixed cumulants. I am still quite sure that I learned a lot and was inspired by the papers of Rota, but maybe this was more about multiplicative functions, and cumulants did not show up there explicitly. At this point I decided to ask Jonathan whether he has some clearer idea about the origin of the classical moment-cumulant formulas. Here is what he said:

About your question, I remember also having a difficult time tracking down a proof of the equivalence of independence and mixed cumulants vanishing in the literature.  I actually think that the earliest paper where this statement is explicitly made, with a complete proof given, is “Cumulants and partition lattices,” by T.P. Speed, Australian Journal of Statistics 25 (1983), 378-388. An annotated version of this paper appears in the collected works of Speed, edited by P. McCullagh (Chapter 6 of the volume). I hope this helps; I don’t know an earlier reference.

I was happy with this and more or less forgot about it. But a few days ago the same issue came up after a talk of Philippe Biane in the online seminar on Algebraic and Combinatoiral Perspectives in the Mathematical Sciences. It seems a couple of people are interested in this and could also provide a bit more information on aspects of the origin of moment-cumulant formulas, and maybe cumulants in general. So I thought it might be a good idea to collect here this information and invite others to add possibly some more remarks on this history.

Franz Lehner offered the following insightful remarks:

Here are some considerations concerning “Rota’s approach to cumulants”.

Both in his posthumous paper Rota/Shen: On the combinatorics of cumulants, J. Combin. Theory Ser. A 91 (2000), and in his Fubini lectures Twelve problems in probability no one likes to bring up he talks about the “Rota approach” but always with reference to Speed. So apparently he never published it himself explicitly, although he certainly knew it for a long time. Speed did not prove any new results, but gave elegant lattice theoretic proofs of known results (his notation is a bit messy though).

On the other hand it must be said that Rota did not invent the Möbius function either as he repeatedly mentions in his 1964 paper, but he clearly saw its fundamental importance (and proved some important results). Rota was a bird in the sense of Dyson and without his efforts to systematize and popularize it, the Möbius function would have remained in its oblivious state for yet another generation.

According to Rota, the Möbius function was invented by Weisner in the thirties. Multiplicative functions and the reduced incidence algebra were introduced in Doubilet/Rota/Stanley Foundations of Combinatorial Theory VI: the idea of generating function, 6th Berkeley Symposium on Probability, 1970/71. Cumulants are not mentioned there, but maybe not without reason this paper appeared in a Symposium on Probability. Similarly his 1964 paper on Möbius functions was not probabilistic, yet it appeared in Probability Theory and Related Fields. Again without mentioning cumulants explicitly, probably because “to prevent the length of this paper from growing beyond bounds, we have omitted applications of the theory”.

He just mentions in passing on p.359 that Schützenberger computed the Möbius function of the partition lattice (independently of Frucht and himself). Indeed in the cited paper Contribution aux applications statistiques de la théorie de l’information (Publ. Inst. Statist. Univ. Paris, 3(1-2):3–117, 1954, Thèse d’État) Schützenberger states as a remark on p.24 the Möbius formula for cumulants. To my knowledge this is the earliest occurrence; Leonov and Shiryaev also use the partition formalism in their 1959 paper, but apparently don’t know the concept of Möbius inversion.

In the statistics literature these developments went largely unnoticed for a long time and the graph theoretic calculation rules of Fisher, Kendall, James etc, remained the tool of choice, see the foreword by McCullagh to Speed’s collected works.

Joachim Kock added the following:

In Kendall’s ‘Advanced Theory of Statistics’ from 1945, there is already an explicit formula for cumulants in terms of moments, and the Möbius function (-1)^{n-1} (n-1)! appears explicitly in the formula! But of course, he doesn’t know that this combinatorial factor is the Möbius function.

In the notes he attributes various moment-cumulant relations to Frisch’s PhD thesis (Oslo 1926), but I don’t know if this particular formula is in there.

Regarding the Möbius function for posets, Weisner’s paper is from 1935 but it only deals with complete lattices, whereas Hall (independently) has the Möbius function for finite posets in his 1936 paper. In both cases, their proofs actually work the same for locally finite posets, which is Rota’s level of generality. (Stretching it a little bit, it is actually the same arguments that work for Möbius categories, and for certain abstract coalgebras.)

In case you are in a historical mood, allow me to advertise my paper From Möbius inversion to renormalisation. (It has no cumulants, though.)

Referring to the combinatorial factor, Franz could add some more insights:

Yes, this expression is already in Thiele’s 1899 paper (reprinted in Anders Hald The Early History of the Cumulants and the Gram-Charlier Series, International Statistical Review 68 (2000) 137-153, in English), but of course not realized as Möbius function, because that one was not known before Schützenberger.

It remains to clarify who was the first to explicitly write moments as a sum over set partitions. Leonov & Shiryaev just infer it from the factorial formula without comment and Schützenberger simply says “nous supposerons connu le fait”.

Thanks to Franz and Joachim for their remarks. Any more comments are welcome …

# Another online seminar: Wales MPPM Zoom Seminar

At the moment there are many online activities going on …. and here is another one: the Wales Mathematical Physics Zoom Seminar, organized by Edwin BeggsDavid EvansGwion Evans,Rolf GohmTim Porter.

Why do I mention in particular this one; there are at least two reasons. Today there is a talk by Mikael Rordam around the Connes embedding problem, and next week I will give a talk, on my joint work with Tobias Mai and Sheng Yin of the last years around rational functions of random matrices and operators.

If you are interested in any of this, here is the website of the seminar, where you can find more information.

Update: The talks are usually recorded and posted on a youtube channel. There you can find my talk on “Random Matrices and Their Limits”.

# Is there an impact of a negative solution to Connes’ embedding problem on free probability?

There is an exciting new development on Connes’ embedding problem. The recent preprint MIP*=RE by Ji, Natarajan, Vidick, Wright, Yuen claims to have solved the problem to the negative via a negative answer to Tsirelson’s problem via the relation to decision problems on the class MIP* of languages that can be decided by a classical verifier interacting with multiple all powerful quantum provers. I have to say that I don’t really understand what all this is about – but in any case there is quite some excitement about this and there seems to be a good chance that Connes’ problem might have a negative solution. To get some idea about the excitement around this, you might have look on the blogs of Scott Aaronson or of Gil Kalai. At the operator front I have not yet seen much discussion, but it might be that we still have to get over our bafflement.

Anyhow, there is now a realistic chance that there are type II factors which are not embeddable and this raises the question (among many others) what this means for free probability. I was asked this by a couple of people and as I did not have a really satisfying answer I want to think a bit more seriously about this. At the moment my answer is just: Okay, we have our two different approaches to free entropy and a negative solution to Connes embedding problem means that they cannot always agree. This is because we always have for the non-microstates free entropy $\chi^*$ that $\chi^*(x_1+\sqrt\epsilon s_n,\dots,x_n+\sqrt\epsilon s_n)>-\infty$, if $s_1,\dots,s_n$ are free semicircular variables which are free from $x_1,\dots,x_n$. The same property for the microstates free entropy $\chi$, however, would imply that $x_1,\dots,x_n$ have microstates, i.e., the von Neumann algebra generated by $x_1,\dots,x_n$ is embeddable; see these notes of Shlyakhtenko.

But does this mean more then just saying that there are some von Neumann algebras for which we don’t have microstates but for which the non-microstates approach give some more interesting information, or is there more to it? I don’t know, but hopefully I will come back with more thoughts on this soon.

Of course, everybody is invited to share more information or thoughts on this!

# 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.