energy scales. As the energy scale or the bin width increases, the values of $D_m(\epsilon)$ monotonically increase. Finally, at about $\epsilon\simeq 100d$ the values of $D_m(\epsilon)$ converges to unity, which indicates that at large energy scales, the strength function appears smooth. The most important feature in Fig. \[fig\_pat\_ldim\] (b) is that the local scaling dimension for $^{208}$Pb almost follows the GOE line at almost all the energy scales.

This should be contrasted with the case of $^{40}$Ca [@aiba2]: The partition function and the local scaling dimension for $^{40}$Ca are shown in Fig. \[fig\_pat\_ldim\] (c) and (d), respectively, for a comparison. When the energy scale is small, the local scaling dimension almost follows the GOE line. As the energy scale increases, however, we can find a dip and a deviation from the GOE line at about 1.7 MeV (Note that $d=12$ keV for $^{40}$Ca). We verified that an occurrence of the dip is not due to a statistical error. Moreover, further studies indicate that the energy where the minimum is located is approximately related to the value of the spreading width of 1p1h states.

Note that if we look only at the small energy scale limit or large energy scale limit, 