Random versus Realistic Interactions for Low-Lying Nuclear Spectra

We compare the shell-model results for realistic interactions with those obtained for various ensembles of random matrix elements. We show that, although the quantum numbers of the ground states in the even-even nuclei have a high probability ( $\sim 60\%$) to be $J^{\pi }{T=0\mathrm{}}^{+}0$, the overlap of those states with the realistic wave functions is very small in average. The transition probabilities $B\left(E2\mathrm{}\right)$ predicted with random interactions are also too small. The presence of the regular pairing is shown to be a significant element of realistic physics not reproduced by random interactions.