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@@ -402,6 +402,15 @@ $\epsilon$ is varied. } \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 |