19 problems

The Ginibre ensemble is a non-Hermitian random matrix ensemble whose eigenvalues and eigenvectors have coupled dynamics; the hidden dynamics of eigenvectors refers to the eigenvect…

Strong non-Hermiticity scaling conjecture. For $cmu<1$, the $cmu$-dependence of the strong-non-Hermiticity microscopic spectral density can be restored by replacing

Consider the ensemble of non-hermitian matrices constructed as $\phi_P$ according to, with $P$, $U$, and $D$ drawn independently, where $P$ is drawn from, $U$ is Haar-distributed o…

Let $k$ be a positive integer and let … where, for every $i,j=1,\dots,k$, $f_{i,j}\in L^{\infty}([-i,i]])$ is real-valued almost everywhere, and either $T_n(f_{i,j})$ is Hermitian…

Let $\vb{A}$ be a large deterministic matrix and $\vb{B}$ a rotationally invariant random matrix. Define the matrix-valued transforms $\mathcal{G}_{\vb{A}+\vb{B}}$,…

Spectral-boundary completeness conjecture. These constructions generate all spectral boundaries of $\vb{M}$. The conjecture concerns whether the available determinations of the tra…

Let $\mathbf{B}$ be a large rotationally invariant random matrix, with $\mathfrak{g}_{1,\mathbf{B}}$ and $\mathfrak{g}_{2,\mathbf{B}}$ denoting the corresponding transform componen…

Let $\mathbf{A}$ and $\mathbf{B}$ be two large independent random matrices, and let $\mathbf{U}$ be a Haar-distributed unitary random matrix. For large $z\in\mathbb{C}$ or large…

Let $T$ be the matrix whose Fourier-mode blocks are $T_k=\sigma\tau\tilde{A}_k\tilde{B}_k$, where the commuting tridiagonal matrices $\tilde{A}_k$ and $\tilde{B}_k$ are … with…

Let $A$ be a random matrix with a doubly stochastic variance profile, meaning that its expected squared column and row norms satisfy … and … For a complex number $z$, write…

Let $T$ be the nilpotent Jordan block with ones on the superdiagonal, let $\gamma\geq 5$, let $n\geq\gamma^2$, and let $R$ be an arbitrary deterministic $n\times n$ matrix whose en…

Let $A$ be an $n\times n$ matrix with i.i.d. subgaussian entries of zero mean and unit second absolute moment. For a unit eigenvector ${\bf v}=(v_1,\ldots,v_n)$, write $v_i^*$ for…

A biunitarily invariant random ensemble is an ensemble of matrices whose distribution is invariant under left and right multiplication by independent unitary matrices. Let…

Microscopic eigenvector universality conjecture. For generic complex non-Hermitian matrices, at every such bulk point there exists the limit

Let $B_n$ be an $n\times n$ matrix with independent and identically distributed entries of zero mean and unit variance, and let the empirical spectral distribution of…

Hidden eigenvector dynamics conjecture. The hidden dynamics of eigenvectors discovered by the authors and described for the Gaussian non-Hermitian ensemble is a general feature of…

Let $\mathbf{X}$ be a non-Hermitian random matrix model whose mean spectral density is rotationally symmetric around zero. For each centered circular component $R=R_{\textrm{b}}$ o…

Let $X$ be a non-Hermitian matrix whose spectrum is rotationally symmetric, and consider the Hermitian matrices obtained by symmetrization and antisymmetrization, namely…

Rotational-spectrum projection conjecture. The projection onto the real axis satisfies