Bohigas–Giannoni–Schmit conjecture for chaotic spectral statistics

Let a classical system be chaotic, and consider its quantum eigenvalues on the scale of the mean level spacing. Let $\lambda$ denote the spectral parameter tending to infinity. The relevant random-matrix ensemble is determined by the system's symmetries, including time-reversal invariance and spin.

Bohigas–Giannoni–Schmit conjecture. The eigenvalues are distributed like the eigenvalues of Hermitian random matrices, with the following limiting ensembles: without time-reversal invariance, the Gaussian Unitary Ensemble (GUE), characterised by quadratic level repulsion; with time-reversal invariance and integer spin, the Gaussian Orthogonal Ensemble (GOE), characterised by linear level repulsion; and with time-reversal invariance and half-integer spin, the Gaussian Symplectic Ensemble (GSE), characterised by quartic level repulsion.

This conjecture is the random-matrix counterpart to the Poisson-statistics prediction for integrable systems. The source states the symmetry-dependent alternatives but gives no resolution, so the conjecture remains open here.