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?