How can I show that $1/N$ expansion for large $N$ matrix models have a string theoretical perturbation expansion?

Question

While surfing through some further reading suggestions on string theory, I stumbled upon this slide from a talk by Nathan Seiberg. I wanted to derive the main argument by applying a perturbation expansion for this partition function, but I couldn't get my head around the concept of a phase space for $N$ by $N$ matrices. I am accustomed to the techniques of path integrals in QFT, but for this case I couldn't even move my pen. How can I learn more about matrix models and partition functions for matrices so that I can show that simple matrix models have a stringy perturbation expansion?

Answer 1

The mentioned expansion of $\ln\mathcal{Z}$ is one of the basic statements for matrix models. If you are familiar with QFT, it is quite easy to investigate matrix models with help of the following notes/books/papers:

1. "Random Matrices" by Eynard et. al. It is well-written book with many examples and useful facts. You can briefly review Chapter 1 and can find the mentioned expansion in Chapter 2.
2. "Matrix Models and 2D String Theory". Here you can find the similar narration but with (in my opinion) more details about strings.
3. "2D Gravity and Random matrices". It is the big and comprehensive book about matrix models and its relations to 2D gravity. You can use it as the handbook.
4. "Applications of Random Matrices in Physics". Finally, if you are intereted about different applications of matrix models in physics (for instance, QCD), you can check this book.

If it is needed, I can reproduce the derivation of free energy expansion, but I assume that references are more useful.

Hope this helps.