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Your matrix model can be written as the product of a standard gaussian matrix with a deterministic diagonal matrix. As such, I believe that Theorem 4 of this paper of Bordenave: http://arxiv.org/abs/ …

There is certainly no asymptotic independence between $\det M_{11}, \det M_{22}$. From the base times height formula for parallelepipeds we see that \begin{align*} \frac{|\det M_{12}|}{|\det M_{22}|} …

I did a preliminary feasibility analysis of our methods and it appears possible that one may be able to tighten our $n^\epsilon$ loss to something more like $\exp( \sqrt{n} )$ in the Gaussian case, bu …

This is studied in Gérard Ben Arous and Paul Bourgade, Extreme gaps between eigenvalues of random matrices, Ann. Probab. 41 (2013), no. 4, 2648--2681. (Ah, so that's how the "insert citation" butto …

The distribution of the bulk of the spectrum is an example of the circular law. For the model you selected (where each entry is uniformly chosen at random from an interval), the law was first proven …

It is a little more convenient to work with random (-1,+1) matrices. A little bit of Gaussian elimination shows that the determinant of a random n x n (-1,+1) matrix is $2^{n-1}$ times the determinan …

I don't know of a fully intuitive derivation, but there are some informal arguments that give the circular law with a relatively small amount of calculation. Let $M$ be a matrix where the entries are …

One does not need to have $n^{-B-3/2}/2$ to be equal to $0.1$, it is enough for it to be less than or equal to $0.1$, which is certainly the case for $n$ large enough. Thanks for pointing out this ty …

I think I can get an upper bound of $O(1/n^2)$ by exhibiting a vector $v$ of magnitude comparable to $1$ which gets mapped to a vector of magnitude $O(1/n^2)$. The basic idea is to exploit the birthd …

By the Chernoff bound, we see that for each $1 \leq i < j \leq m$, one has $u_i \cdot u_j = O(\sqrt{n})$ with probability at least $1-\frac{1}{10m^2}$ (say), where implied constants are allowed to dep …

One can get a certain way towards this goal via a sort of "dimensional analysis". This isn't a completely satisfying heuristic argument - in particular, it only partially specifies what compression m …

Firstly, the equation you attribute to Silverstein (and is sometimes known as the "self-consistent equation" for the Stieltjes transform) is not exact, but only asymptotically valid in the limit $n \t …

We have $$ C(W) = 2 A \circ A + v v^\top$$ where $v$ is the vector with entries $\|w_i\|^2$, $A$ is the Wishart matrix with entries $w_i^\top w_j$, and $\circ$ is the Hadamard product. From the Schur …

The continuous comparison method of Knowles and Yin, Knowles, Antti; Yin, Jun, Anisotropic local laws for random matrices, Probab. Theory Relat. Fields 169, No. 1-2, 257-352 (2017). ZBL1382.15051. fol …