Merge branch 'master' of https://git.overleaf.com/5fcce4736e7f601ffb7e1484

1 files changed, 5 insertions, 5 deletions

diff --git a/bezout.tex b/bezout.tex
index 1a12025..b0ead4e 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -328,14 +328,14 @@ Consider for example the ground-state energy for given $a$, that is, the energy
   } \label{fig:complexity}
 \end{figure}
 
-\textcolor{teal}{ {\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a range of values of $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$.
+{\color{teal} {\bf somewhere} In Figure \ref{fig:desert} we show that for $\kappa<1$ there is always a gap of  $a$ close to one for which there are no solutions: this is natural, given that the $y$ contribution to the volume shrinks to zero as that of an $N$-dimensional sphere $\sim(a-1)^N$.
 For the case $K=1$ -- i.e. the analytic continuation of the usual real computation -- the situation
-is more interesting. In the range of  values of $\Re H_0$ where there are real solutions there are solutions
-all the way down to $a=1$: this is only possible if the density of solutions diverges at this value: this is natural, since. 
+is more interesting. In the range of  values of $\Re H_0$ where there are exactly real solutions this gap closes, and this is only possible if the density of solutions diverges at $a=1$. 
+Another remarkable feature of the limit $\kappa=1$ is that there is still a gap without solutions around
+`deep' real energies where there is no real solution. A moment's thought tells us that this is a necessity: otherwise a small perturbation of the $J$'s could produce a real, unusually deep solution for the real problem, in a region where we expect this not to happen.
 }
 
-
-\begin{figure}[htpb]\label{desert}
+\begin{figure}[htpb]
   \centering
   \includegraphics{fig/desert.pdf}
   \caption{