15 problems

Fix $alpha0$. Let $mu1leqcdotsleqmuN$ be the eigenvalues of a GOE matrix, and let $lambda1leqcdotsleqlambdaN$ be the eigenvalues of a real generalized Wigner matrix $(HN)$ with…

Let $(H_N)$ be a generalized Wigner ensemble with uniform $p$-support, and let $mu1leqcdotsleqmuN$ be the eigenvalues of a GOE matrix while $lambda1leqcdotsleqlambdaN$ are the ei…

Let $A_N$ be a finite-rank diagonal matrix as in Theorem 1.1, and let $\mathbf{u}_1,\ldots,\mathbf{u}_N$ be eigenvectors of a real symmetric Wigner matrix. The theorem's extremal-s…

Let $\mathbf{W}_n$ be a large Wigner matrix, let $H^{\mathbf{W}_n}$ denote its eigenvector empirical spectral distribution, and let $F$ be the semicircle law. The VESD and the ESD…

Let $\mathbf{W}_n$ be a large Wigner matrix with eigenmatrix $\mathbf{U}_n$. For a unit vector $\mathbf{x}_n\in\mathbb{C}^n$, write $\mathbf{y}_n=\mathbf{U}_n^*\mathbf{x}_n$. Motiv…

Let $M_n$ be a Wigner matrix, let $i$ lie in the bulk region ${\varepsilon} n \leq i \leq (1-{\varepsilon}) n$, and let $u=u_{i/n}$ be the classical location determined by … Let…

Tao–Vu eigenvalue rigidity conjecture. The factor $N^{-1/6-\varepsilon_0}$ on the right-hand side should be replaced by $N^{-2/3+\varepsilon}$.

Twelfth-moment conjecture. The existence of the twelfth moment

Twelfth-moment conjecture. The existence of ${\bf E}\,\lvert a_{ij}\rvert^{12}$ should be necessary for the existence of a universal upper bound for $M_{2s_n}^{(n)}$ as…

Bulk eigenvalue asymptotic conjecture. One has

Let $H$ be a universal Wigner matrix with spread parameter $M$, and let $\mathbf{v}$ be an $\ell^2$-normalized eigenvector of $H$. The available result gives delocalization on a sc…

Let $\lambda_j$ be the eigenvalues of a standard Wigner matrix, and let $\gamma_j$ denote their classical locations. Under the assumptions of the cited estimate, the root-mean-squa…

Let ${\bf M}_N$ be a deformed Wigner matrix whose perturbation has $k=k(N)$ distinguished directions, with $1\ll k\ll \sqrt{N}$ in the proved results. Rank-growth conjecture. The s…

Consider a deformed GOE matrix $M_N^G$ with a finite-rank deterministic perturbation $A_N$. In the rank-one case, take $A_N$ with diagonal form $D_N={\rm diag}(\theta,0,\ldots,0)$…

Let $(M_N)_N$ be complex or real deformed Wigner matrices formed from Wigner matrices whose entry distribution is $\mu$ and from deterministic finite-rank perturbations $A_N$. In t…