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If $U$ is a $N\times N$ random unitary matrix uniformly distributed with respect to Haar measure, a $M\times M$ block $A$ from it has distribution given by $$ \det(1-AA^\dagger)^{N-2M}.$$ If $O$ is a …

I am interested in matrix integrals, and I have seen many mentions to a certain Riemann-Hilbert approach that indicate that this is a very powerful tool to can be used in this area, when coupled with …

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 …

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 …

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 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 …

Given two lists $i$ and $j$ of $2n$ positive integers less than $N$, Collins and Sniady have computed, in Integration with respect to the Haar measure on unitary, orthogonal and symplectic group (see …

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 …

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 …

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 …

The diagonal elements of $P=\frac{1}{N}MM^T$, like $$P_{11}=\frac{1}{N}\sum_{i=1}^NX_i^2,$$ satisfy $ \langle P_{11}\rangle=1$ and $ \langle P_{11}^2\rangle=1+2/N$ (variance decreases like $N^{-1}$). …

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 …

The Oxford handbook of random matrix theory (Oxford University Press, 2011), edited by G. Akemann, J. Baik, P. Di Francesco, is an excellent reference, which covers a wide variety of properties and ap …

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 …

I am writing more than a year after the question was posted, only to spell out some more details regarding the calculation implicit in the solution presented by Suvrit, and to clarify the dependence o …