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authorJaron Kent-Dobias <jaron@kent-dobias.com>2021-02-08 15:09:30 +0100
committerJaron Kent-Dobias <jaron@kent-dobias.com>2021-02-08 15:09:30 +0100
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Added paragraph describing typical norm of critical points.
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} \label{fig:desert}
\end{figure}
+In the thermodynamic limit \eqref{eq:complexity.zero.energy} implies that most
+critical points are concentrated at infinite $a$, i.e., at complex vectors with
+very large squared norm. For finite $N$ the expectation value $\langle
+a\rangle$ is likewise finite. By differentiating $\overline{\mathcal N}$ with
+respect to $a$ and normalizing, one has an approximation for the distribution
+of critical points as a function of $a$. The expectation value this yields is
+$\langle a\rangle\propto N^{1/2}+O(N^{-1/2})$. One therefore expects typical
+critical points to have a norm that grows modestly with system size.
+
These qualitative features carry over to nonzero $\epsilon$. In
Fig.~\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. When $\kappa=1$---the analytic