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My initial answer was wrong. Instead of completely deleting it, I left it at the bottom of the post. I use notation now that slightly differ from my original post. As before, I give only a partial a …

I assume your random $M_i$s are from $O(3)$, not $SO(3)$. In fact, I suspect that the answer depends on whether $M_1M_2$ has eigenvalue $1$ or eigenvalue $-1$. If I did not make a mistake, when you e …

You did not specify whether J is assumed symmetric or not. If it is, the limit of empirical values of eigenvalues of J alone is the semicircle, while in the non-symmetric it is the circular law. If …

For $p>1$, the random variables you discuss do not possess exponential moments; You are in the regime of large deviations with stretched exponential tails. See for example the following recent paper b …

Expanding a bit on Yemon Choi's comment: concentration is indeed the key. First, $\|B\|_F^2$ is simply the sum of the squares of singular values of $A$, and $E\|B\|_F^2=m(m-1)n+mn^2$. On the other ha …

All you need is that the empirical measure of the $k_i$s converge. Whenever it does, just apply the original Pastur-Marcenko (1967) results to get that the empirical measure associated with your matr …

In the Gaussian case, you can rewrite $x_i=R^{1/2}y_i$ where $y_i$ now possess iid entries. This leads you to computing the eigenvalues of $Y^*RY$, this is actually the problem solved by Pastur and Ma …

To amplify on Brendan's answer: You are dealing with a matrix of zero mean iid's plus a rank one perturbation: $M=W+ A$ where $A=c11^T$ and $c=EY$. The spectral norm of $W$ is asymptotically $2\sqrt …

Let $X$ be the array after your operation. The law of $X_{i,j}$ for $(i,j)$ chosen in a deterministic way from $[1,..,n]^2$ converges to the uniform law. The tilting due to your operations is very sma …

I think the following answers your question (and more): http://www-personal.umich.edu/~romanv/papers/product-random-deterministic.pdf

An explicit formula is the hook-product formula, due to Schensted I believe. This formula is used in the classical work of Logan and Shepp, as well as in Vershik-Kerov. See for example equation (1.1) …

Actually, just to get the semicircle is not hard. Take an $n$-by-$n$ Jacobi matrix whose on diagonal entries are $0$ and $i$th entry on the off diagonal is $\sqrt{i/n}$. The limit ESD will be the semi …

Intuitively, the fact that the density vanishes at the edge already hints that the spacing at the edge changes (which indeed it does), and that the asymptotics are different; this is confirmed in the …

There are many ways to see that, maybe the simplest is the following: let $W=XX^T/K$. Compute $E Tr W^k$ as $K,N\to\infty$. The combinatorics is easy - essentially, the only terms that survive passag …

The relation between singular values and eigenvalues in the invariant case is evaluated in this paper, under some technical conditions: http://arxiv.org/abs/0909.2214 (see http://annals.math.princeton …