
Free Probability Theory
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A blog on freeing probability from its commutative chains. This is a blog on topics around Free Probability Theory. I plan to post here all kind of information which is relevant in the context of free probability theory, like: meetings, new results, discussions of open problems or general directions in the subject.
Free Probability Theory
2M ago
In the post “Structured random matrices and cyclic cumulants: A free probability approach” by Bernard and Hruza I mentioned the problem about the effect of non-linear functions on random matrices and showed some plots from my ongoing joint work with Alexander Wendel. Finally, a few days ago, we uploaded the preprint to the arXiv. What ..read more
Free Probability Theory
8M ago
My paper with Johannes and Tobias on the noncommutative Edmonds’ problem has got a total makeover; the new revised version appeared just on the arXiv. It relies now heavily on the recent results of Tobias and mine on the Fuglede-Kadison determinant of matrix-valued semicircular elements. But also, the organization of the paper is now much more reader-friendly – at least we hope so. Everything you need to know can be gotten from the first three sections; if you need proofs, read the rest.
Let me first recall the noncommutative Edmonds’ problem. Consider a matrix
,
where the are noncommuting va ..read more
Free Probability Theory
8M ago
Consider a matrix with integer entries, like
How can we convince ourselves that there are no two eigenvalues close to zero; say, you should convince me that two eigenvalues both with absolute value smaller than 0.1 are not possible. Of course, we should not calculate the eigenvalues but decide on this by arguments as soft as possible.
It is easy to get upper bounds for eigenvalues by the operator norm or the normalized trace of the matrix – those are quantities which are usually easy to control.
Not so clear are lower bounds, but actually it’s the determinant which will be of good use here. N ..read more
Free Probability Theory
1y ago
The evolution of some papers of Bernard and Hruza in the context of “Quantum Symmetric Simple Exclusion Process” (QSSEP) have been addressed before in some posts, here and here. Now there is a new version of the latest preprint, with the more precise title “Structured random matrices and cyclic cumulants : A free probability approach“. I think the connection of the considered class of random matrices with our free probability tools (in particular, the operator-valued ones) is getting nicer and nicer. One of the new additions in this version is the proof that applying non-linear functions entry ..read more
Free Probability Theory
1y ago
Voiculescu’s online seminar on probabilistic operator algebra is running again, for the spring term, but now with a different time; it’s not on Mondays anymore, but on Tuesdays. For more info and details about the upcoming talks, see here. In particular, to get the zoom links for the talks, one should write to Jorge (jgarzav_at_caltech_dot_edu).
Actually, the next talk, on January 30 at 10:30 a.m. Pacific time (which is 7:30 p.m. German time), will be given by myself. It will be on Bi-free probability and reflection positivity and is based on my small note on the arXiv. Most of the talk will b ..read more
Free Probability Theory
1y ago
In the recent post of a similar title I mentioned some papers which related physics problems (eigenstate thermalization hypothesis or Open Quantum SSEP) with free probability. Let me point out that the title of the preprint by Hruza and Bernard has been changed to “Coherent Fluctuations in Noisy Mesoscopic Systems, the Open Quantum SSEP and Free Probability” and that there are some new and follow up preprints in this directions, namely “Spectrum of subblocks of structured random matrices: A free probability approach“, by Bernard and Hruza, and also “Designs via free probability“, by Fava, Kurc ..read more
Free Probability Theory
2y ago
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 ..read more
Free Probability Theory
2y ago
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” qu ..read more
Free Probability Theory
2y ago
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 gen ..read more
Free Probability Theory
2y ago
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 ..read more