Nearest Level Spacing Statistics in Open Chaotic Systems: Generalization of the Wigner Surmise

We investigate the nearest level spacing statistics of open chaotic wave systems. To this end we derive the spacing distributions for the three Wigner ensembles in the one-channel case. The theoretical results give a clear physical meaning of the modifications on the spacing distributions produced by the coupling to the environment. Based on the analytical expressions obtained, we then propose general expressions of the spacing distributions for any number of channels, valid from weak to strong coupling. The latter expressions contain one free parameter. The surmise is successfully compared with numerical simulations of non-Hermitian random matrices and with experimental data obtained with a lossy electromagnetic chaotic cavity.

Figure 1

Distribution of the normalized spacings for the 3 Gaussian ensembles with $⟨\Gamma ⟩/\Delta =1$. On the top for GOE, on the middle for GUE, and on the bottom for GSE. On the left for $M=1$ and on the right for $M=10$. For the one-channel case, the exact analytical expressions (5, 8, 9) are represented; for $M=10$, considering $1/\eta$ as a free parameter. In the inset the tail of the distributions is represented in semilog scale. For all curves, the mean level spacing is normalized to one.

Figure 2

Evolution of the effective coupling strength $1/{\eta }_{\mathrm{eff}}$ as a function of the normalized mean width $⟨\Gamma ⟩/\Delta$ for the GOE case. $M=1$ in solid line, $M=3$ in dots, and $M=5$ in dashed line.

Figure 3

Distribution of the spacings for an open chaotic microwave cavity. In histogram is the experimental distribution, on the top with the range 1 to 6 GHz ($M_w=10$) and on the bottom 15 to 16 GHz ($M_w=20$). Both histograms have been obtained using 2600 events. The dotted line shows the Wigner surmise, and the solid line the best fit using expression 5. The confidence level is lower than 50% for the Wigner surmise (1) and larger than 90% for the expression (5).