Non combinatorial random matrix theory

Question

I am learning random matrix theory. Unfortunately I do not like combinatorics, and have never really been good at it. But I found that random matrix theory has heavily relied on combinatorics, particularly in finding the limiting spectral distribution, at least for symmetric matrices, where the moment method is the most popular one. It may have an elementary appeal but I do not really find any beauty in that and I am also aware of its limitations.

I am a probabilist. Unfortunately in all the problems in random matrix theory I have seen, there is almost zero probability and a lot of counting involving graphs and trees. It is not doing any justice to my knowledge of probability.

I am studying from Bai and Silverstein's book "Spectral Analysis of Large Dimensional Random Matrices". It is a good book in terms of the results it has collected and some interesting ways in which it has proved them. But then I have been terribly disappointed in the combinatorial way it has approached the subject.

I am looking for questions in random matrix theory that are not combinatorial in nature. I like analysis a lot more, so if you can suggest that I look into an area that has both random matrix theory and analysis, I would be thrilled. I feel that I am not getting any insight while applying these counting principles, in that I am just solving a problem but not really understanding the structure of a random matrix. I would like to understand a random matrix. Please help me find an answer.

I searched online but was surprised there is no result on non combinatorial random matrix theory. It is strange that everybody studying random matrix theory is very happy with the combinatorial jargon thrown at them!

NOTE: I am aware of an invariance principle developed by Chatterjee, although I have not studied that in depth. In case you are aware of that, does it provide a different perspective into the universality of random matrices?

Answer 1

Two successful techniques for obtaining the limiting spectral measure of large Hermitian random matrices are (i) the moment method and (ii) the Stieltjes transform method. The moment method is indeed very combinatorial.

The Stieltjes transform method, however, does not involve any combinatorics. The key idea is to derive a self-consistent equation for the normalized trace of the resolvent $$m(z)=\frac{1}{n}\text{Tr}\, (H-z)^{-1}= \frac{1}{n}\sum_i \frac{1}{\lambda_i-z}$$ for Hermitian $n\times n$ matrices $H$ with eigenvalues $\lambda_1,\dots,\lambda_n$ and $z$ in the upper half plane. Using either the Schur complement formula, or more probabilistic techniques like Stein's method, one can show that $m(z)$ approximately satisfies some self-consistent equation. In the case of Wigner matrices, for example, one finds $$m(z)\approx -\frac{1}{m(z)+z}.$$ Solving this quadratic equation gives the Stieltjes transform of the semicircular distribution, of course.

Also note that in the spectral bulk the Stieltjes transform method is more powerful than the moment method. More precisely, $m(x+iy)$ contains local information about the eigenvalues close to $x$ on a scale of $y$. Such local information is very difficult to extract from the moments.

Answer 2

I think this text by Eynard, Kimura and Ribault may interest you. There are some of those diagrams in chapter 2, but there are also nice connections to algebraic geometry, loop equations and integrable systems.

Answer 3

The connection between the spectral edge of random matrices and integrability theory (Painlevé systems) may well be a topic that appeals to your interest in analysis. Here is a talk on Random Matrix Theory and the Askey Table-Painlevé-Sakai Scheme that summarises some of the issues, and here is a paper on Application of the τ-function theory of Painlevé equations to random matrices that could be helpful as an entry point to the literature.

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If I interpret the OP in a more restricted sense as asking for a non-combinatorial proof of the eigenvalue density of an ensemble of large random matrices, then I could refer to A Note on the Eigenvalue Density of Random Matrices for a probabilistic proof using a variational principle.
Or alternatively, Random Matrices with Complex Gaussian Entries is explicitly written with the motivation that: "Many results on eigenvalue distributions for random matrices are obtained by complicated combinatorial methods, and the purpose of this paper is to give more easily accessible proofs, by analytic methods, for those results on random matrices, which are of most interest to people working in operator algebra theory and free probability theory".