I was reading Alice Guionnet's book "Large Random Matrices: Lectures on Macroscopic Analysis". I would need some help in understanding the author intends to do in Part III, "Matrix Models".
In the earlier chapters, she talks about the proof of Wigner's semi circle law and joint convergence along with Voiculescu's theorem. Then she discusses large deviation inequalities for largest eigenvalue of Wigner matrices, and also derives finer estimates of the moments of Wigner matrix.
The chapters included in this part are "Maps and Gaussian Calculus", "First Order Expansion" and "Second Order Expansion for Free Energy". In the preface the author mentions that this section deals with Gaussian matrix integrals in a perturbative regime.
I am quite thrown off by the technicalities in this section, and I believe the general philosophy of what one does with a matrix integral in not clear to me. I understand that a matrix integral is related to combinatorics of maps but if one can explain in simpler words what is going on there, how the logic flows, that is the basic skeletal steps, I would be immensely thankful.
But then, for a beginner in random matrix theory, would it be prudent to dwell on this? For instance, one would not suggest large deviation inequalities immediately to a beginner in probability theory. I have a feeling of importance for the Wigner law, but not really for the matrix integrals, which seem a bit esoteric to me.
So my question is twofold: Can you please give a basic idea of what exactly the author is doing in this section of the book? Is this important enough to be studied thoroughly or should it be left for a later reading by a specialist?
