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I'll post an answer to spell out all the details. You have $$G(ω)=\frac{1}{N}E\left[{\rm Tr}\frac{1}{Iω−J}\right]=\frac{1}{N}E\left[\sum_\lambda\frac{1}{ω−\lambda}\right]$$ This can be written as $$ …

The case $k=n$ is a consequence of the identity $$\int \det(f_j(s_k))\det(g_j(s_k))\prod_{j=1}^N d\mu(s_j) = N!\ \det\left(\int d\mu(t) f_j(t)g_k(t)\right)$$ which I have seen under the names "And …

The spectrum of random matrices has some universality properties, in the sense that they do not depend on details of the distribution of the matrix elements. For instance, the spectral gap, i.e. the s …

It is known that the partition function $$ \mathcal{Z}_1=\int dH e^{-N{\rm Tr}(H^2)}e^{-NV(H)},$$ where the integral is over $N\times N$ hermitian matrices $H$, with the potential $$ V(H)=\sum_{j\ge 1 …

Just expanding a bit on the comment by Beenakker. If you write the Gaussian measure in terms of the matrix elements, $$ \prod_{ij}\exp[−\frac{N}{2(1−τ^2)}(J_{ij}^2-\tau J_{ij}J_{ji})]$$ You can see …

The probability that a $n\times n$ real matrix (with elements that are independent random variables with standard normal distributions) has only real eigenvalues is given by $$ 2^{-n(n-1)/4}$$ Refere …

A good source on this topic is chapter 7 of Roob Muirhead's book, "Aspects of Multivariate Statistical Theory", or chapter 3 of Peter Forrester's "Log-Gases and Random Matrices". In particular, there …

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 integrabl …

The book "The Drunkard's Walk: How Randomness Rules Our Lives" by Leonard Mlodinow is probably close to what you are looking for.

The crucial point is that the eigenvectors are not unique, as David Handelman commented. There is a gauge freedom since you can change their phase at will. If you want the map $U\to(D,V)$ to be biject …

If $A$ is a Gaussian random matrix as you describe, then the ensemble of matrices given by $A^TA$ is known as the Wishart ensemble, or the Laguerre ensemble. It has been extensively studied, and you c …

I think the following papers are closely related to what you are looking for: How many eigenvalues of a Gaussian random matrix are positive? (2011) Index Distribution of Gaussian Random Matrices (20 …

I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgo …

Complementing the answer by Carlo, if you take all $k$'s equal you have $$f_{\rm GUE(d)}(k,...,k)\propto \int dX e^{ik{\rm Tr}(X)}e^{-\frac{d}{2}{\rm Tr}(X^2)}.$$ Taking $x$ to be any real diagonal el …

The joint probability distribution of the eigenvalues of $S$ is proportional to $$ \rho(S)=e^{-{\rm Tr}(S)}\det(S)^{a/2}|\Delta(S)|,$$ where $a=m-d-1$ and $\Delta(S)$ is the Vandermonde. The average v …