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There is few posts on MO that asked about reference on this topic, and I found some difficulty during the process of getting myself into the subject so here is the question.

I really want to hear from someone who is familiar with this wonderful field of random matrices.

1 I would like to know a self-study material on the subject of random matrix theory, which should be more advanced than [Tao]. I did read into [Tao]'s "Related article" part but found it focus on dynamics instead of a general interest like the content of [Mehta] covered.

2 About the classic in the field [Mehta], I am confused about its editions since some probabilist said the 2nd edition is better than the third one while 3rd edition is almost 200 pages of more than the 2nd edition.

Since I have not started to read it yet, I would like to know which edition of [Mehta] is better to start (as a newcomer to the subject) with, and what is the difference between these two editions(besides those newly added references Mehta mentioned in the 3rd edition's preface, which is not very informative to me...)?

3 Maybe this should be another post, but how much (statistical) mechanics should I know if I want to read [Deift&Gioev] in order to understand it better? Since my basic interest is in the mathematical side, is there any good-and-short introductory paper providing an overview about the subject of statistical mechanics?

4 Lastly, I would really like to know if there is some must-read or introductory paper on the subject of random matrices. (Besides [Tao].)

And a roadmap of learning this subject, if possible, will be greatly appreciated.

Reference

[Mehta]Mehta, Madan Lal. Random matrices. Vol. 142. Academic press, 2004.

[Deift&Gioev]Deift, Percy, and Dimitri Gioev. Random matrix theory: invariant ensembles and universality. Vol. 18. American Mathematical Soc., 2009.

[Tao]Tao, Terence. Topics in random matrix theory. Vol. 132. Providence, RI: American Mathematical Society, 2012. https://terrytao.files.wordpress.com/2011/02/matrix-book.pdf

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    $\begingroup$ What did you end up buying Henry? $\endgroup$ – arjun Nov 9 '17 at 23:02
  • $\begingroup$ @arjun I bought Mehta's 2nd edition, which is cheaper. I am happy with it and getting something out of it. If you are interested, you can see some of my other answers about random matrices. Good luck! $\endgroup$ – Henry.L Nov 25 '17 at 0:14
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The Oxford handbook of random matrix theory (Oxford University Press, 2011), edited by G. Akemann, J. Baik, P. Di Francesco, is an excellent reference, which covers a wide variety of properties and applications of random matrices (this is a very diverse subject). It is not a textbook, but a collection of introductory papers by different authors, which are well written and have many references that you can follow up.

I must also mention the book Log-gases and random matrices, by Peter Forrester (Princeton University Press), which is probably close to what you want (advanced, comprehensive, pedagogical).

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  • $\begingroup$ Hi, Marcel. Thank you and I will look into it. BTW, Are you familiar with [Mehta]? Do you use the handbook as a reference or learning book since it has 900 pages? $\endgroup$ – Henry.L Feb 25 '17 at 3:08
  • $\begingroup$ I think all these books (I have the third edition of Mehta) can be used both for learning and as references. They are long and have both introductory sections and result-packed sections. I have updated my answer to include another source. Hopefully you will get more answers. $\endgroup$ – Marcel Feb 26 '17 at 2:35
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The notes based on lectures by Bertrand Eynard (https://arxiv.org/pdf/1510.04430.pdf) will serve as a nice self-contained text for Random Matrix Theory. (These lecture notes discuss three different approaches to random matrix models and have very stimulating discussions on related mathematical ideas.) And I guess these notes can (rather, should) be supplemented with Mehta's book.

Regarding your second question, I have no idea how the second and third editions of Mehta's Random Matrices compare with each other. I have just read chapters from the third edition and never got to look into the second edition of the book.

Again, for the third question, I do not know if there is any good-and-short review from where you can learn all the statistical mechanics one needs to understand random matrix texts. I think that you will anyway pick things up while going through standard random matrix texts.

Now, to answer your fourth question, one paper I'd surely suggest you to read is Freeman Dyson's The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics (http://aip.scitation.org/doi/10.1063/1.1703863). It is one of the earliest papers introducing the ideas that are now central to the theory of random matrices.

As for a road map, my understanding is that the texts you should be referring to afterwards will almost completely depend on what topics you're more interested in/want to pursue reading further. There are quite a few articles available online which specifically discuss particular aspects/applications of random matrix theory.

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