Good introduction to NCP and free probability

Question

I am new to the topic of random matrix theory (RMT) and am looking to learn about non-commutative probability spaces and some of the applications to RMT.

Do you know good lecture notes, treatments, or books on the subject?

I came across a unital functional $\varphi$ which in RMT functions a bit like a trace (the expected average trace to be precise) and saw this defined in conjunction with commutative and "free" random variables and non-commutative ones. These terms, including "free probability" are all new to me and I am looking to learn these techniques.

Thank you so much for your help!

Answer 1

These are the resources I could gather in the form of an answer. Some of these I have not personally read but have found "enough" positive reviews (both online and from professors). Firstly, a very popular text (also mentioned in a comment above):

• Free Probability and Random Matrices by James Mingo and Roland Speicher. A very good introduction to the subject of Free Probability with solutions to exercises at the back.
• Terrence Tao's Topics in Random Matrix Theory based on his Graduate Topics course on the subject.
• You can also check out Roland Speicher's blogsite on Free Probability Theory for a bunch of resources.
• Terry Tao also has a blogsite on Random Matrices.
• I have also found the book Random Matrices and Non-Commutative Probability by Arup Bose helpful.

Additionally, I came across these two "short" (lecture) notes:

• Introduction to non-commutative Probability by Isak Mottelson.
• Non-commutative Probability Theory by Paul D. Mitchener.

Hope these resources would help you at least get a nice introduction to the subject.