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Diffstat (limited to 'bezout.tex')
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1 files changed, 31 insertions, 1 deletions
@@ -131,7 +131,7 @@ singular values of $\partial\partial H$, while both are the same as the distribution of square-rooted eigenvalues of $(\partial\partial H)^\dagger\partial\partial H$. -{\color{red} {\bf perhaps not here} This expression is to be averaged over the $J$'s as +{\color{red} {\bf perhaps not here} This expression \eqref{eq:complex.kac-rice} is to be averaged over the $J$'s as $N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation that involves the replica trick. In most, but not all, of the parameter-space that we shall study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{ \mathcal N_J} = \ln \int dJ \; N_J$ is exact. @@ -167,6 +167,36 @@ where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue spectrum of $\partial\partial H$ therefore is that of an ellipse whose center is shifted by $p\epsilon$. +\begin{figure} + \centering + + \raisebox{60pt}{$|\epsilon|=0$} + \hfill + \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_0.0.pdf}} + \hspace{2pt} + \includegraphics{fig/spectra_singular_0.0.pdf}\\ + \raisebox{60pt}{$|\epsilon|=\frac12|\epsilon_{\mathrm{th}}|$} + \hfill + \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_0.5.pdf}} + \hspace{2pt} + \includegraphics{fig/spectra_singular_0.5.pdf}\\ + \raisebox{60pt}{$|\epsilon|=|\epsilon_{\mathrm{th}}|$} + \hfill + \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_1.0.pdf}} + \hspace{2pt} + \includegraphics{fig/spectra_singular_1.0.pdf}\\ + \raisebox{60pt}{$|\epsilon|=\frac32|\epsilon_{\mathrm{th}}|$} + \hfill + \raisebox{5pt}{\includegraphics[scale=1.0]{fig/spectra_eigenvalue_1.5.pdf}} + \hspace{2pt} + \includegraphics{fig/spectra_singular_1.5.pdf} + + \caption{ + Eigenvalue and singular value spectra for the matrix $\partial\partial H$ + for $a=\frac54$, $\kappa=\frac34e^{i3\pi/4}$, and $\epsilon=i|\epsilon|$ for values values of $|\epsilon|$. + } \label{fig:spectra} +\end{figure} + The eigenvalue spectrum of the Hessian of the real part, or equivalently the eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is the singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the |