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@@ -406,15 +406,16 @@ marginal minima are isolated from each other.
\caption{
The neighborhood of marginal states at several energies below the threshold
energy. \textbf{Left:} The range of energies $E_1$ at which nearby states
- are found. For any $E_0<E_\mathrm{th}$, the nearest class of states is at a
- nonzero distance, and their energies are higher than that of the reference
- configuration. \textbf{Center:} The range of stabilities $\mu_1$ at which
- nearby states are found. As below, there is always a sufficiently large
- overlap beyond which all nearby states are saddle with an extensive number
- of downward directions. \textbf{Right:} The range of energies at which
- \emph{other} marginal states are found. Here, the more darkly shaded
- regions denote where an isolated eigenvalue appears. Marginal states above
- the threshold are always separated by a gap in their overlap.
+ are found. For any $E_0<E_\mathrm{th}$, the nearest class of states is at
+ an extensive distance, and their energies are higher than that of the
+ reference configuration. \textbf{Center:} The range of stabilities $\mu_1$
+ at which nearby states are found. For $E_0$ near the threshold, the nearest
+ states are always index-one saddles with $\mu>\mu_\mathrm m$, but as the
+ overlap gap widens their population becomes model-dependent.
+ \textbf{Right:} The range of energies at which \emph{other} marginal states
+ are found. Here, the more darkly shaded regions denote where an isolated
+ eigenvalue appears. Marginal states belo the threshold are always
+ separated by a gap in their overlap.
} \label{fig:marginal.prop.above}
\end{figure}