The results are shown in the uppermost panel of Fig. \[fig:mom01\]. The moments as a function of $\Delta/J$ all fall on the same curve in this case. If the hopping parameters $J$ and $J'$ are switched (not shown), the sign of the odd moments changes, the even moments remain the same. These results are in accordance with section \[sec:FDL\].

Fig. \[fig:mom01\] also shows the cumulants for different ratios; $J'/J=0.3, 0.5, 0.7$. The deviation of the cumulants from one another is more pronounced, and increases with an increase of $J'/J$. However, the moments become equal for any $J'/J$ when $\Delta \rightarrow \pm \infty $. In this case also, the system becomes an independent array of two state systems. The band energies in all these cases vary continuously with $k$ across the Brillouin zone.

The results for the case $J'/J=1$ are also shown separately in Fig. \[fig:mom1\], as well as the limits $J' \rightarrow_\pm J$. For finite $\Delta/J$ the odd cumulants are zero, indicating an even probability distribution. The ratio of the second and fourth cumulants rule out a Gaussian. As $\Delta/J \rightarrow 0$ a discontinuity in the slope of the band develops. In this case, the cumulants $C_2$ and $C_4$ diverge. 