\section{Higgs-to-peaks-assignment with and without correlations}
In order to determine the best-fitting Higgs boson combination for a peak, we calculate for all possible Higgs boson combinations their $\chi^2$ contribution for this observable. Then, we select the Higgs combination which gives the least $\chi^2$ and fulfills certain conditions\footnote{If the difference between the observed peak mass and the mass of the Higgs boson is less than the (gaussian combined) mass uncertainty, we require this Higgs boson to be assigned to a peak found in the analysis (which is typically only one)}. In this $\chi^2$ calculation, we do not take into account correlations with other analyses and their Higgs boson assignments, for the following reason: The covariance matrices for the $\chi^2$ contribution from the Higgs mass depend on the assignment
of Higgs bosons to the peaks due to the theoretical mass uncertainties. Thus, the correlation between two peaks highly depends whether they share the same assigned Higgs boson or not. Thus, if we wanted to take into account correlations to find the best (global) Higgs-to-peaks-assignment, we would have to go through \textit{every possible Higgs-to-peaks assignment (for all peaks)} and then \textit{find the least total $\chi^2$ (from all peaks)}. This would be computationally expensive (but maybe necessary, as you can see below).
Now, the following may happen: Two different Higgs combinations may have very similar $\chi^2$ (\textit{without correlations}) contributions. Thus, slight changes of the input parameters may lead to a change of the Higgs-to-peaks assignment. However, the true $\chi^2$ contribution (\textit{with correlations}) might be quite different. Thus, in total $\chi^2$ returned by \HS~might have a jump at this transition of Higgs-to-peak assignments.
An example is shown in Fig.~\ref{Fig:heavyH}. Here, all three neutral Higgs bosons of the model are possibly assigned to peaks, so there are many transitions in this plane. Comparing the plot without correlations (a) with the one with correlations (b), we see that there two structures (around $(m_A,\tan\beta)\approx(115,3)~\mbox{and}~(140,3)$) in (b) where the total-$\chi^2$ makes a jump to higher values. The total $\chi^2$ without correlations shown in (a) has a much smoother shape. Note, that the Higgs-to-peak assignment for every parameter point is identical for both (a) and (b).