Generalized random matrix conjecture for chaotic systems

Abstract: The eigenvalues of quantum chaotic systems have been conjectured to follow, in the large energy limit, the statistical distribution of eigenvalues of random ensembles of matrices of size $N\rightarrow\infty$. Here we provide semiclassical arguments that extend the validity of this correspondence to finite energies. We conjecture that the spectrum of a generic fully chaotic system without time-reversal symmetry has, around some large but finite energy $E$, the same statistical properties as the Circular Unitary Ensemble of random matrices of dimension $N_{\rm eff} = \tH / \sqrt{24 d_1}$, where $\tH$ is Heisenberg time and $\sqrt{d_1}$ is a characteristic classical time, both evaluated at energy $E$. A corresponding conjecture is also made for chaotic maps.