I am learning random matrix theory . I am aware that the most popular successful techniques for obtaining the limiting spectral measure of large Hermitian random matrices are the moment method and the Stieltjes transform method. The moment method is indeed very combinatorial and the Stieltjes transform method is analytical nature. The key idea is to derive, by using analytical tools, a selfconsistent equation ($m(z)\approx \frac{1}{m(z)+z}$) for the normalized trace of the resolvent $$m(z)=\frac{1}{n}\text{Tr}\, (Hz)^{1}= \frac{1}{n}\sum_i \frac{1}{\lambda_iz}$$ for Hermitian $n\times n$ matrices $H$ with eigenvalues $\lambda_1,\dots,\lambda_n$ and $z$ in the upper half plane. The Stieltjes transform method is also used in Wireless telecommunication for dealing with deterministc equivalents, witch are roughly speaking matrix combinations. The Stieltjes transform method is generally preferred over the moment method since it considers the eigenvalue distribution of large dimensional random matrices as the central object of study, while the moment approach is dedicated to the specific study of the successive moments of the distribution. Note in particular that not all distributions have moments of all orders, and for those that do have moments of all orders, not all are uniquely defined by the series of their moments. My questions is: What are the disadvantages of the moment method over the resolvent method in Random Matrix Theory?

1$\begingroup$ the large$z$ Taylor expansion of $m(z)$ gives you the individual moments, so the Stieltjes transform can be seen as the generating function of the moments; so basically your question is what are the advantages/disadvantages of studying individual moments over their generating function; it all depends on what you need, if knowledge of mean and variance is sufficient, you don't need the full generating function. I don't think there is much more to say. $\endgroup$– Carlo BeenakkerJun 30, 2018 at 14:56

$\begingroup$ I think the moment methods is more power to deal with moments than the Stieljes transform. My question is to know for which random matrix models the Stieljes transform is so poor than the moment method. $\endgroup$– IliyoJul 1, 2018 at 10:19
1 Answer
The Stieltjes transform method is better for getting quantitative estimates on the empirical spectral distribution at small scales. In particular, it is the method by which the local semicircle law was established. See this survey of BenaychGeorges and Knowles: https://arxiv.org/abs/1601.04055. The advantages of the resolvent/Stieltjes transform approach are discussed on pages 4–5.

$\begingroup$ When the moment method is better than the Stieltjes transform? $\endgroup$– IliyoDec 21, 2018 at 23:48