14 problems

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…

The quantum mechanical system is desymmetrized with respect to all its unitary symmetries, eigenvalues are considered on the scale of the mean level spacing, and generic degeneraci…

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 $\gamma\in(0,2)$, let $\mu_\gamma$ be the Liouville measure on $D$, and let $(\boldsymbol{\lambda}_j)_{j\ge1}$ be the LQG eigenvalues. Write $F_{\mathrm{GOE}}(x)$ for the GOE l…

Let $X_1,\ldots,X_L$ be independently (not necessarily identically) distributed random matrices with heavy tails and unitary invariance, meaning that their eigenvalues and eigenvec…

Neumann surplus central limit conjecture. Then

Let $H$ be the random matrix and let $\beta$ and the statistic $\mathcal{A}_N$ be as in Theorem. Let $Z_k$, $k\geqslant 1$, denote the random variables appearing in that theorem, a…

For each asset-price time series, form its trajectory matrix by the maximal trajectory-matrix construction used in the source, and consider the matrix's spectral distribution and s…

Let $z_0$ be a point in the bulk of the pseudospectrum, and let $\tilde G_{z_0}$ be the determinant of the $J\times J$ matrix of independent Gaussian analytic functions arising as…

Let $A$ be a random real symmetric banded matrix of size $N\times N$ with independent entries and bandwidth $b\asymp N^\alpha$. Poisson/Gaudin–Mehta conjecture. The limiting local…

Let ${\mathcal{Q}}(\bm \phi)$ be the ensemble of matrices considered in the paper, with eigenvalues $\lambda_i(\bm \phi)$, and let $\rho(x)$ denote their local eigenvalue density.…

Let ${\mathcal L}_0$ be a Diophantine irrational lattice. Write $n_j$ for the distinct Laplacian eigenvalues, let $\delta_j=n_j-n_{j-1}>0$, let … and define the mean-normalized spa…

Bohigas–Giannoni–Schmit conjecture. If the classical problem is a sufficiently chaotic system, then the local statistics of the $E_n$ are generically those of EGO or EGU, according…

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