# Tagged Questions

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### What is known about the distribution of eigenvectors of random matrices?

Let $A$ be a real asymmetric $n \times n$ matrix with i.i.d. random, zero-mean elements. What results, if any, are there for the eigenvectors of $A$? In particular: How are individual eigenvectors ...
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### Moments of the trace of orthogonal matrices

Let $O_n$ be the (real) orthogonal group of $n$ by $n$ matrices. I am interested in the following sequence which showed up in a calculation I was doing $$a_k = \int_{O_n} (\text{Tr } X)^k dX$$ where ...
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### Reference request: results on the asymptotic distribution of singular values related to a random orthogonal matrix

Let $Q$ be a random variable taking as its values the set of $n \times k$ real matrices with orthogonal columns, and whose distribution is the Haar measure on the Stiefel manifold $O(n)/O(n-k)$. This ...
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### Estimating the probability that $\|Av\| \ge \|v\|$

Given a diagonalizable matrix $A \in \mathbb{R}^{n \times n}$ with real eigenvalues, satisfying $1+c_1 \le \rho(A) \le 1+c_2$ $(0<c_1 \le c_2)$, obviously there exists a $v \in \mathbb{R}^{n}$ such ...
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### Probability a random Toeplitz matrix is singular

Consider Toeplitz matrices where the entries in the first row and column (which define the whole matrix) are independently chosen to be either $1$ or $0$ with probability $1/2$. Define $p_n$ to be the ...
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### “Ergodicity” for eigenvalues of random matrices?

Sorry if the wording of this question is sloppy, I have a weak background in probability theory (hence the quotation marks throughout). Is there some "ergodicity-type" result for Wigner's semicircle ...
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### Quasi-classical limit of representation theory

I am looking for a good reference on a general phenomenon of quasi-classical limit in representation theory, which relates "large" representations to measures on (co-adjoint orbits of) the associated ...
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### What is the spectral norm of a random projection times a diagonal?

Take $n\ll N$. Let $P$ be an $n\times N$ matrix of iid $\mathcal{N}(0,1)$ random variables, and let $D$ be an $N\times N$ diagonal matrix. What can be said about the distribution of the largest ...
It is a remarkable fact that the union of congrent cubes has only at most near-quadratic combinatorial complexity, $O^*(n^2)$ for $n$ cubes, known to be almost tight. This contrasts with the union of ...