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this is a limit of a more general result by Majumdar and company, How many eigenvalues of a Gaussian random matrix are positive? (2010), see also their earlier papers from 2006 and 2008. The coeffici …

You want to think of the Haar measure $d\mu(U)$ as a way of measuring uniformity in the group $U(N)$ of unitary $N\times N$ matrices. To form your intuition, consider $N=1$. You then have $U=e^{i\phi} …

If you choose the matrix elements of $A$ independently from a Gaussian distribution you have the socalled Ginibre ensemble of random-matrix theory. The statistics of the eigenvalues is known, see for …

Pastur and Vasilchuk have extended the result of Diaconis and Evans for $a_{2k}$ from $2k\leq n/2$ to $2k\leq n-1$: $$a_{2k}=\pi^{-1/2}2^{k}\Gamma(k+1/2)\;\;\text{for}\;\;2k\leq n-1\quad\quad[ …

Involutions $s=s^{-1}$ in $S_n$ are modeled by the Tracy-Widom distribution $F_1$ for real symmetric matrices (GOE): Take as $S_n^\ast$ the subset of involutions in $S_n$, and let $M_n$ be the corresp …

Indeed, the distribution function of the eigenphases of a random matrix in $\operatorname{SO}(n)$ has a peak at 0 and at $\pm\pi$. It only becomes uniform for large $n$. The joint distribution functio …

A recursion formula for the moments of the Gaussian orthogonal ensemble, M. Ledoux (2009). The desired recursion formula for the moment $b_p^N\equiv E\,[\,{\rm tr}\,(S_N^{2p})]$ is I notice a diffe …

The answer to the question as stated (maximum of row elements) has been solved in Extreme statistics of complex random and quantum chaotic states, see also this MO posting: $$\int dU \max_j |U_{1,j} …

Here are some applications of free probability of random matrices: Neural networks: The asymptotic freeness assumption plays a fundamental role in the study of the propagation of spectral distributio …

This next-nearest-neighbor distribution of the Riemann zero's is addressed in Mehta's book on random-matrix theory. It is well reproduced by that of the Gaussian Unitary Ensemble (GUE), compare black …

$$\int_{{\rm U}(n)} dU\,|{\rm Tr}\,(U^m)|^2={\rm min}\,(n,m).$$ see Theorem 2.1.b of Diaconis and Evans (2001). [*] [*] This 2001 reference corrects an earlier paper by Diaconis and Shahshahani (199 …

The entire set of correspondences can be read off from this table: Listed are the 10 symmetric spaces and for each space in the left column the dual space is shown in the right column, as explained …

High-Dimensional Probability, An Introduction with Applications in Data Science, by Roman Vershynin (draft version freely available) The two texts by Van Handel and Vershynin are compared here: R …

Since $O$ is orthogonal, $O^\top=O^{-1}$ commutes with $O$, hence the eigenvalues $\mu_n$ of $O+O^\top$ are related to the eigenvalues $e^{i\phi_n}$ of $O$ by $\mu_n=2\cos\phi_n$. The spectral density …

The Distribution of the Determinant of a Complex Wishart Distributed Matrix proves that the determinant is distributed as the product of independent random variables with a chi-squared distribution, …