13 problems

Let $M=N$ and consider the special case $\beta=2$ for the Wishart ensemble. Denote the minimum eigenvalue by $\lambda_{\rm min}$ and expectation by $\langle\cdot\rangle$. Znidaric'…

Wishart Gaussian product inequality conjecture.

Determinantal formula. There is a function $C(x)$ such that

Wishart eigenvector bead-process convergence conjecture. The coupled eigenvalue--eigenvector dynamics of Wishart matrices are expected to converge to the same limiting Markov chain…

Let $x_{11}$ be an entry of the underlying data matrix, and write $4\xi=\operatorname{Var}(x_{11}^2)4$. Let $T_{n,p,m}$ denote the maximum largest eigenvalue over the relevant…

Let $\mathfrak{X}$ be the Wishart random matrix in the setting of the cited theorem, partitioned into diagonal blocks $\mathfrak{X}_{ii}$ of widths $p_1,\ldots,p_d$. Wishart princi…

Let $p\in\mathbb{N}$, let $\alpha\in(p-1,\infty)$, and let $\Sigma\in\mathcal{S}_{++}^p$. Let $\mathfrak{X}\sim\mathcal{W}_p(\alpha,\Sigma)$ be a $p\times p$ Wishart random matrix,…

Let $\alpha$ be the anisotropy parameter, let $X$ be as in the definition of the anisotropic Wishart model, and let $W^*(n,\alpha)$ denote the law of … Let $M^*(n)$ be the law of a…

Universality conjecture. Under these assumptions, the asymptotic behavior of

Asymptotic eigenvalue-dominance conjecture. This relation holds for arbitrary $m$ and arbitrary configurations of $\Sigma$ and $\Omega$.

Let $W=(NQ)^{-1}$ be the Wishart matrix described above, and define $c_0(\mu_1,\dots,\mu_\ell)$ by … Here $\ell\geq 1$, $(\mu_1,\dots,\mu_\ell)\in\mathbb{N}^{\ell}$, and $C_\ell$ d…

Let $W=XX^*$, where $X$ is an $N\times(N+\alpha)$ matrix of independent standard complex normal random variables, with … where $1/2<\gamma\leq1$, and let $\nu=4N+2\alpha+2$. The es…

First-correction conjecture. By inspection of the Painlevé V equation, the first correction to the limiting form is conjectured to have the form