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| -rw-r--r-- | 2-point.tex | 19 |
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diff --git a/2-point.tex b/2-point.tex index 7c7d345..4e14f4c 100644 --- a/2-point.tex +++ b/2-point.tex @@ -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} |