17 problems

A random matrix ensemble is classified by its symmetry class: real symmetric, complex Hermitian, or quaternion. Local eigenvalue statistics, including eigenvalue gaps and other sta…

For each $N$, choose uniformly from the finite set of $N\times N$ adjacency matrices of simple $d$-regular graphs, and consider their normalized eigenvalue spacings. GOE spacing co…

Let $p$ be a probability distribution used to sample independent entries of real symmetric random matrices, and consider the normalized eigenvalue spacings of the resulting matrice…

Let $A_N$ be an $N\times N$ real symmetric Toeplitz matrix from the Toeplitz ensemble, and consider its eigenvalues after normalization so that the local spacings between adjacent…

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…

Eigenvalue-variance conjecture. As $L\to\infty$,

Most stably singular initial-condition conjecture. Among all initial conditions for Dyson Brownian motion, those maximizing $R^{(1)}_\beta(0)$ are precisely

Delocalization and universality conjecture. The eigenfunctions should be completely delocalized, and the local eigenvalue statistics should remain in the Wigner–Dyson–Mehta univers…

Let $Z\in\mathcal{G}(n,p)$, let $\lambda_k$ denote its ordered eigenvalues, let $q_k$ be the corresponding classical locations, and write $\tilde{k}=k\wedge(n+1-k)$. Assume…

Let $U$ be an $N\times N$ random matrix from the circular unitary ensemble with eigenvalues $\{e^{i\theta_j}\}_{1\le j\le N}$, where $0\le \theta_1\le\dots\le\theta_N<2\pi$. Choose…

Let $M$ be the real symmetric band random matrix defined by independent centered entries with variance $(1+\delta_{jk})\sigma^2$, normalized by $b_n^{-1/2}$, and band radius $b_n$.…

Let a random matrix ensemble be specified by a probability distribution on a set of matrices, and consider the local statistics of its eigenvalues after the appropriate microscopic…

Universality conjecture. The marginal densities $p^{(n)}_{l,\beta}$ in this scaling limit should be universal, meaning that they do not depend on the potential $V$. This conjecture…

Consider ensembles of $N\times N$ real symmetric matrices whose entries are independently sampled from a probability distribution indexed by $p$, and normalize their eigenvalues so…

Local eigenvalue statistics of random-matrix ensembles should be determined by the symmetries of the ensembles, but otherwise be independent of the details of the distributions. Un…

Let a Wigner random hermitian matrix ensemble have extreme eigenvalues, meaning its minimal and maximal eigenvalues, in the large matrix limit $n\to\infty$. Asymptotic independence…

Let the one-dimensional Anderson model have eigenvalues (EVs), and let $I$ be an interval. Poisson statistics conjecture. The distribution of the distances between eigenvalues come…