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Last month, I worked on a literature review project on "Random matrices and Wigner's Semi-circle law" as part of my course. During the project I happened to encounter a new concept called "free random variables" and the associated theory of "Free probability" as being a non-commutative analogue of the Classical Probability. I found the topic to be quite interesting from the little I understood about it (I read the introduction given in Terence Tao's book Topics in Random Matrix Theory).

So to understand more, I tried to read Dr. Roland Speicher's notes on Free probability but couldn't comprehend the Math involved. I have not yet been exposed to the topics that were used in those notes.

Hence I am putting this question here. I want to learn more about free probability and am looking for a text that talks in the context of random variables and not operator algebras (those notes involved this topic, but I do not know anything about operator algebras. By the context of random variables, I mean the way Free probability is introduced in Tao's book as something that discards information about probability spaces and concerns with the space of random variables alone). I don't have much knowledge about the subject, so I don't know if my request makes sense or not. So it will also be helpful if you can guide me about how should I approach this subject and what necessary background I will require to learn it.

(I am a Master's student in Statistics in India. I have undertaken semester courses in measure theoretic probability, real analysis and linear algebra. And I am comfortable with these topics.)

P.S. There are no existing tags on Free probability and I cannot create one. So I am putting a tag of Random matrices because it being the area where I encountered this topic.

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    $\begingroup$ I don't know much about free probability, but a fine introduction to operator algebras for probabilists can be found over at the almost sure blog (almostsure.wordpress.com/2020/02/05/…). $\endgroup$
    – WoolierThanThou
    Aug 25, 2020 at 15:21
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    $\begingroup$ Or well... I guess that story sort of starts here (almostsure.wordpress.com/2019/11/10/algebraic-probability). $\endgroup$
    – WoolierThanThou
    Aug 25, 2020 at 15:22
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    $\begingroup$ I'm a masters student in statistics (in India) too, and while I quite liked the approach taken up by Nica-Speicher, I totally understand your qualms as you described. In a similar course, our professor started the approach from a more combinatorial way, and his LaTeX file is quite comprehensive (but not public for me to share), but afaik, you have to pick up $C^*-$algebras for the study sooner or later. Maybe look at lectures $1,2,8-12$ before you feel comfortable enough. $\endgroup$
    – Fawkes4494d3
    Aug 25, 2020 at 16:28
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    $\begingroup$ I suppose you could try looking through some general topology and functional analysis (added with your experience from measure theoretic probability) so that the occurrence of "algebra"-like structures (like a topological space, or a measure space) becomes more regular and familiar to you. The blog mentioned by @WoolierThanThou is a good one, I follow it too. You can check that out, and update yourself with definitions as you go along. Somewhere, you have to take up the study of hardcore analysis. (Our prof gave us the warning that the Speicher text was a pure math intro, so .... ~\0_0/~) $\endgroup$
    – Fawkes4494d3
    Aug 25, 2020 at 16:35
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    $\begingroup$ @SanketAgrawal , I'm not sure, maybe you could do that. It'd be great if you could. This is his homepage. $\endgroup$
    – Fawkes4494d3
    Aug 25, 2020 at 17:11

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I am of course quite biased here, but it is good to get any feedback on the Speicher notes. I have written quite a few texts on free probability, on different kind of levels. Maybe you should have a look on my survey article Free Probability Theory and its avatars in representation theory, random matrices, and operator algebras; also featuring: non-commutative distributions And there is also a recorded lecture series on free probability which I gave a few years ago.

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  • $\begingroup$ Wow! An answer from the man himself. Sorry I am late to see this post but I will definitely check out these links. Thanks. $\endgroup$
    – Sanket Agrawal
    Oct 15, 2020 at 17:47
  • $\begingroup$ @RolandSpeicher: May I ask you how you did typeset your visualization of the kernel of a partition in your lecture notes? (The result consists of lines like in your profile image.) $\endgroup$
    – user7427029
    Mar 9, 2021 at 1:33
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    $\begingroup$ @user7427029: nothing fancy, just using tikzpicture; here is an example \begin{tikzpicture}[baseline=-15pt] \begin{scope}[xshift=2.2em, scale=0.8] \draw (0,0em) -- ++ (0,-2em) -- ++(4em,0) -- ++(0,2em); \draw (4em,-2em) -| ++(2em,2em); \draw (5em,0) -- ++ (0,-1em); \draw (1em,0) -- ++ (0,-1.5em) -| (3em,0); \draw (2em,0) -- ++ (0,-1em); \end{scope}; \end{tikzpicture} $\endgroup$
    – Roland Speicher
    Mar 21, 2021 at 11:57
  • $\begingroup$ Thank you very much! $\endgroup$
    – user7427029
    Mar 21, 2021 at 21:43

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