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Statistics of spectral properties of matrix-valued random variables.

0 votes

Non-asymptotic results for bulk of random Wishart matrix

I don't have time for details right now, but you want to look at circa 1990 papers by Alan Edelman and Stanislaw Szarek (independently) on condition numbers of random matrices. For the state of the a …
Mark Meckes's user avatar
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1 vote

Least singular value gaussian orthogonal ensemble.

I'm not sure offhand (and don't have time to check at the moment) if the GOE version of this is known, but the distribution least singular value of a nonsymmetric $n \times n$ matrix with i.i.d. norma …
Mark Meckes's user avatar
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3 votes
Accepted

Induced p-norm of a Random matrix

Here are some papers you might want to look at if you're interested in this issue (probably better than the paper of mine that Suvrit mentioned): G. Bennett, V. Goodman, and C.M. Newman. Norms of ra …
Mark Meckes's user avatar
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2 votes

Random complex eigenvalues and averages of traces

Carlo's answer addresses the first question. To address the second one: the "disk law" (better known as the "circular law") does not tell you the distribution of singular values. However, the Marchen …
Mark Meckes's user avatar
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2 votes

Distribution of trace of inverse-Wishart matrix $W_n(I,n)$

Assuming I'm interpreting your notation correctly, with high probability it is known that $\lambda_{\min}(W_n) \ge c/n$ for some absolute constant $c$ (see Edelman, "Eigenvalues and condition numbers …
Mark Meckes's user avatar
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3 votes

Does the Tracy-Widom distribution describe the tails of eigenvalue densities of finite dimen...

As Terry pointed out in a comment, the TW law is only valid asymptotically. However, there are various results which show that the asymptotics for finite dimension, which TW heuristically suggest, ar …
Mark Meckes's user avatar
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2 votes
Accepted

Extending GUE to a measure on operators?

One way of looking at this is that, up to normalization, the GUE is standard Gaussian measure on the space of Hermitian matrices, equipped with the Hilbert-Schmidt or Frobenius inner product. So what …
Mark Meckes's user avatar
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2 votes
Accepted

concentration of sums of fourth moment of normals

What counts as "best"? The smallest tail bound is of course $$ (2\pi)^{-n/2} \int_{\{(x_1,\dotsc,x_n) : \sum x_i^4 > (1+t)3n\} } e^{-\sum x_i^2 / 2} dx_1 \dotsb dx_n. $$ Presumably you want somethin …
Mark Meckes's user avatar
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9 votes

Derandomizing random matrices

There is active interest in such results in high-dimensional geometry, and expander graphs have even been used explicitly as a tool. Take a look for example at this paper and the references on the se …
Mark Meckes's user avatar
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9 votes

Statistics for Haar measure of random matrices?

I'm guessing that you meant to write "real orthogonal matrices" without "symmetric", since the set of symmetric orthogonal matrices has Haar measure 0. Otherwise, please clarify what you mean by "a d …
Mark Meckes's user avatar
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5 votes
Accepted

Expected value of the spectral norm of a Wishart matrix?

I don't think there's an exact expression, but the Bai–Yin result does give the right prediction. It's a little easier to state nice-looking results for a $p \times n$ matrix $X$ with independent sta …
Mark Meckes's user avatar
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1 vote

Spectra of VERY sparse random matrices

Here's just one quick remark (a little involved for a comment) about how you can start making greater use of your suggested reduction to $A$. Since $M-A$ has rank 1, $$ \sigma_3(A) \le \sigma_2(M) \le …
Mark Meckes's user avatar
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1 vote
Accepted

Asymptotics of the right singular vectors as the number of rows diverge

Thanks to the rotation-invariance of the Gaussian distribution, both $U_m$ and $V_m$ can be taken to be Haar-distributed orthogonal matrices in $\mathbb{R}^{m\times m}$ and $\mathbb{R}^{n\times n}$ re …
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