99 problems

In quantum chaos, consider a quantum system whose classical dynamics are sufficiently chaotic. Bohigas–Giannoni–Schmit conjecture. Its local spectral statistics should be described…

Generic quantum integrable systems are quantum systems with integrable classical dynamics, and locally Poissonian level statistics means that their unfolded neighbouring energy lev…

Let a quantum system be chaotic, and consider the local features of its spectrum. Random matrix theory (RMT) is the statistical framework relating such spectral features to random…

Wigner-Dyson-Gaudin-Mehta conjecture. The local spectral statistics of Wigner matrices exhibit universality: they are independent of the underlying distribution of the matrix entri…

Let $A$ be the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices, let $H=A/\sqrt{d-1}$, and consider the bulk eigenvalues of $H$. Fixed-degree random-…

Poisson statistics conjecture. If $a^2/b^2$ is a “good” irrational number, then the local spectral statistics of the rectangular billiard are well described by Poisson statistics f…

Let an $N\times N$ random band matrix (RBM) have bandwidth $W$. The eigenvectors may be completely delocalized for large bandwidth and exponentially localized for small bandwidth,…

Let $R_k$ be the $k$-point correlation function of the eigenvalues of an $N\times N$ Hermitian random matrix, let $E$ lie in the bulk of the spectrum, let $\rho(E)$ denote the limi…

Consider quantum many-body systems, such as spin-$1/2$ or fermionic systems, that lack a meaningful classical or semiclassical description. Restrict to sectors in which all local c…

Let $X$ be a fixed compact hyperbolic surface, and let $T,L>0$ satisfy $T\to\infty$, $L\to\infty$, and $L=o(T)$. Denote by $\sigma_{\chi,\psi}^{2}$ the limiting variance appearing…

Gamma-distribution conjecture. The distributions $P_n(s)$ are well described by normalized gamma distributions

Gaussian level-density conjecture. For $N\gg1$, the level density follows a Gaussian distribution.

Energy-statistics correspondence conjecture. A regular classical system corresponds to Poisson energy-level statistics, while an irregular classical system corresponds to Wigner–Dy…

Let $\Delta^{\rm Poisson}(L)$ and $\Delta^{\rm RMT}(L)$ denote the Poisson and random-matrix predictions for the spectral rigidity, and let $q$ be the irregularity fraction of the…

Mehta's universality conjecture. The local statistics of the eigenvalues of $A$ are, in the limit $n\to\infty$, determined only by the overall symmetries of the ensemble and are in…

For a random operator with disorder, consider the energy levels of finite-volume restrictions on the scale of their typical energy spacing. Write “pp” for pure-point spectral regim…

Let $H$ be a quantum Hamiltonian whose classical counterpart is integrable, and consider the point spectrum of $H$ (or its normalized energy differences). Berry's conjecture. The p…

Let $H_N$ be an $N\times N$ Hermitian random matrix with independent entries, not necessarily Gaussian, and let the local spacing distribution mean the distribution of the distance…

Let the classical phase space be the Riemann sphere $\mathbb{C}P^1$, and let the classical map be a fixed-time map $\exp(t\Xi_H)$ of a Hamiltonian flow. Its quantization is a unita…

Calogero–Sutherland mapping conjecture. The spectral statistics of critical PLBRM with large $b$ can be mapped onto those of the Calogero–Sutherland model at low temperature

Let a critical power-law banded random matrix (PLBRM) have variance parameter $b$ in the regime of large $b$. The spectral statistics of the PLBRM are compared with those of fermio…

FHK postulate. The interval counts $\delta\mathcal{N}_I^{({\boldsymbol J})}$ should be log-correlated and, as $\Delta\to\infty$, should converge to a Gaussian random model of the f…

Let $d\geq 2$, and let ${\mathscr L}_N$ be the randomly twisted transfer operator defined using independently uniform permutation matrices. Write $\mathrm{Sp}({\mathscr L}_N)$ for…

Let a random-matrix ensemble have a complex eigenvalue spectrum, and consider its bulk spectral statistics after local rescaling. Haldane–Kausch–Kieburg–Urbina's second conjecture.…

A hyperbolic random graph places vertices in a hyperbolic space, typically $\mathbb{H}^2$, with edges formed according to probabilities that decay with hyperbolic distance; its deg…