43 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…

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

Bulk Wigner eigenvector bead-process conjecture. The sequence $\left(\Theta_N^{(s)},\Upsilon_N^{(s)}\right)_{0\leq s\leq K}$ defines a coupled Markov chain, which converges in dist…

Let $X$ be an $N\times N$ matrix, let $z$ be an eigenvalue, and let ${\bf v}$ and ${\bf u}$ be corresponding right and left eigenvectors. Normalize the right eigenvector by…

Let $\{w_j\}_{j=1}^{p}$ be a deterministic sequence of real values such that $|w_j|\le 1$ for all $j\in[\![p]\!]$. Let $\{\mathbf{v}_j\}_{j=1}^{p}$ be the eigenvectors of $\Sigma$,…

Let a Hermitian random matrix be viewed as a Hamiltonian of a disordered quantum system, and let its eigenvectors be normalized to lie on the unit sphere. Quantum Unique Ergodicity…

Consider the critical regime of the random matrix model, where the interaction strength satisfies $ert Aert_{{\mathrm{HS}}}\sim 1$, and let a bulk eigenvector be decomposed into it…

Eigenvector coordinate conjecture. With high probability,

Eigenvector variance conjecture. With high probability, the eigenvectors of these graphs are delocalized and have a variance approximated by the density of the corresponding cumula…

Let $v_1$ be the eigenvector associated with the largest eigenvalue, and let the variational problem defining the rate function be optimized over the vector $\chz$. Localization-pr…

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…

For each $k\geq 3$, let $\Delta_k$ be the maximum Chebyshev distance between the coordinatewise powers of the principal eigenvectors of a connected $k$-graph and its clique-shadow:…

Let ${\bf B}_n$ be a Bernoulli matrix in the regime where $pn$ is of order one, and consider its eigenvectors and the limiting spectral measure, whose atomic part consists of Dirac…

Let $\mathbf{m}^*$ be the optimal mean-shift vector and let $\mathbf{d}^*_1$ be the leading eigenvector of the matrix $\Sigma^*$, with estimators $\hat{\mathbf{m}}$ and…

Let $\widehat{{{v}}}_2$ and $\widehat{{{u}}}_2$ be the estimated eigenvectors, and let $v_{i2}$ and $u_{i2}$ denote their corresponding population eigenvector entries for each fixe…

Let $G$ be the random graph and let $\nu_1,\dots,\nu_n$ be the eigenvectors appearing in Theorem 1 of the paper, with $p$ as in that theorem. For a vector $v$, write $\|v\|_\infty$…

Anomalous-dimension symmetry conjecture. The anomalous dimensions should satisfy

Kravtsov's fractal-eigenvector conjecture. The eigenvectors are delocalized, but only over $N^{2-\gamma}$ sites close in energy, yielding a fractal dimension for the eigenstates.

Let $g_{ij}$ denote the Gaussian entries of a Wigner matrix, and consider replacing their distribution by any subexponential distribution. Delocalization means that the mass of eac…

Eigenvector generation conjecture. For each $n\geq 1$, a generating set of $S_{\mathbb Q}\langle a,b\rangle_n$, consisting only of eigenvectors, may be obtained by applying all pos…

Expanding orthogonal similarity variant conjecture. For almost all matrices of dimension at most $7$, repeatedly applying this eigenvector construction and random orthogonal or uni…

Let $A$ be a matrix of dimension at most $7$, and repeatedly construct eigenvector matrices $X_i$ whose columns have unit $\mathbb{L}_2$ norm. A matrix property holds for almost al…

Let $X$ be a random matrix satisfying the assumptions of Theorem, and let $T$ be its Schur form. Define $Y=n|t_{12}|^2$, where $t_{12}$ is the relevant off-diagonal entry of $T$, a…

GHD universality conjecture. The GHD can be considered a new universal distribution of extended states, on an equal footing with the PTD.

Mirlin's conjecture. All states in PLBM are extended.