The lower panel in this figure shows what happens when $J'$ is close to $J$ (bigger or smaller) but the two are not quite equal ($J' = J +
  \epsilon$, $\epsilon$ a small number). We see that in this case the first moment is one or minus one, depending on the sign of $\epsilon$, and zero is not approached as $\epsilon \rightarrow 0$ from either side. The left inset in the lower panel shows the behavior of the first moment on the $\Delta-\delta$ plane, indicating a discontinuity along the line $\Delta<0,
  \delta = 0$ (the well-known result of Vanderbilt and King-Smith [@Xiao10; @Vanderbilt93]). The moments and cumulants we find are consistent with the behavior shown in the left inset of the lower panel of the figure.

In Fig. \[fig:prob\] we show examples of reconstructed probability distributions for $J'/J=0.0, 0.3, 0.5, 0.7$, in each case for several values of $\Delta/J$. $\chi^2$ (defined in Eq. (\[eqn:chisquared\])) is tabulated in the appendix (Table \[tab:chi2\]). The most localized example ($J'/J =
0$ and $\Delta/J=-2$) shows a sharp peak around $x=1$; as $\Delta$ decreases the curves shift to the left and spread out, but their shape is always very similar (for smaller values of $\Delta/J$ 