The geometry of multi-parameter families quantum states is important in numerous contexts, including adiabatic or nonadiabatic dynamics, quenches, and the characterization critical points. Here, we discuss Hilbert-space eigenstates parameter-dependent random-matrix ensembles, deriving full probability distribution geometric tensor for Gaussian Unitary Ensemble. Our analytical results give exact...

Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $\phi$ be the characteristic polynomial. Conditionally on extended Riemann hypothesis, with high probability is irreducible $\mathrm{Gal}(\phi) \geq A_n$.