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Suppose $x_i\in \mathbb{R}^d$ are IID isotropic random vectors with $\|x_i\|=1$ and matrix $A_d$ is defined as follows: $$A_d=\prod_i^d \left(I-x_ix_i^T\right)$$ Is anything known about the spectrum …

Terry Tao RMT book has the following formula for joint moment of freely independent random variables $X,Y$ in Section 2.5 $$\tau(XYXY)=\tau(X)^2\tau(Y^2)+\tau(X^2)\tau(Y)^2-\tau(X)^2\tau(Y)^2$$ Exerci …

Suppose $C_n$ is a product of $n$ $d\times d$ matrices with IID entries coming from standard normal. The following appears to be true. Is there an elementary proof? $$E[\|C_n\|_F^2]=d^{n+1}$$ This fol …

Let $A$ be a product of $n$ $d\times d$ matrices with IID standard Gaussian entries and consider the value of $g(x)=x f(x)$ where $f(x)$ is the density of squared singular values of $A/\|A\|$. Is any …

Suppose $M$ is an $n\times n$ matrix with IID random entries drawn from $\mathcal{D}$ and $\sigma$ is the vector of its singular values. Define purity of $M$ as $$\rho(M)=\frac{n \sum_i \sigma_i^4}{\l …

How does the spectrum of a product of $k$ random matrices behave around 0? In particular, I'm wondering if the CDF of squared singular values behaves as $x^{\frac{1}{k+1}}$ around 0. The result for $k …

Suppose $\Sigma=\operatorname{diag}(h)$ where $h=(1^{-p},2^{-p},3^{-p},\ldots,d^{-p})$ and $p> 1$ $X$ is a matrix with $b$ rows sampled independently from $\operatorname{Normal}(0,\Sigma)$ Suppose $d$ …

How can I estimate the following value where $A,B$ are $d\times d$ matrices and expectation is taken over all random permutations of the product? $$E_\text{shuffle}[\operatorname{Tr}\underbrace{AA\cdo …

Suppose $H$ is a random positive definite matrix normalized to have norm 1. Can we say anything about behavior of $f(s)$? $$f(s)=\operatorname{Tr}[H(I-H)^s]$$ Taking $H=A^T A$ with entries of $A$ samp …

Suppose $x_i \stackrel{\text{i.i.d}}{\sim} \mathcal{N}(\mu,\Sigma)$. What can we say about dependence on $b$ of Frobenius/spectral norm quantities below? $$f(b)=\left\|\frac{1}{b}\sum_{i=1}^b x_i x_i^ …

Suppose we sample $k$ vectors $v$ from normal distribution centered at zero and diagonal covariance with diagonal entries $1,\frac{1}{2},\ldots,\frac{1}{d}$ and normalize $v$: $$\frac{v_1}{\|v_1\|},\f …

Typical magnitude of $i$th largest eigenvalue of an Erdos-Renyi random graph seems to decay at least exponentially with $i$. Is there an analytic expression for the constant in the exponent, or a nice …

(also asked on math.se, with no answers) Suppose I have $m$ samples drawn from a Gaussian in $\mathbb{R}^n$, and need sample covariance $\Sigma_m$ to be $\epsilon$-close to true covariance $\Sigma$: $ …

Given a finite set of matrices $A_i$, sample $n$ matrices uniformly with replacement and compute $f_n=\|A_1 A_2\cdots A_n\|^2$. When is the following limit finite? $$\lim_{n\to \infty} E[f_n]$$ I'm es …