From 25891cd9dfd49dc8bb1e23553fafed36419bfe25 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Tue, 8 Dec 2020 18:03:18 +0100 Subject: Spectra figure added. --- bezout.tex | 32 +++++++++++++++++++++++++++++++- fig/spectra_eigenvalue_0.0.pdf | Bin 0 -> 6861 bytes fig/spectra_eigenvalue_0.5.pdf | Bin 0 -> 6865 bytes fig/spectra_eigenvalue_1.0.pdf | Bin 0 -> 6859 bytes fig/spectra_eigenvalue_1.5.pdf | Bin 0 -> 6865 bytes fig/spectra_singular_0.0.pdf | Bin 0 -> 9319 bytes fig/spectra_singular_0.5.pdf | Bin 0 -> 9871 bytes fig/spectra_singular_1.0.pdf | Bin 0 -> 10376 bytes fig/spectra_singular_1.5.pdf | Bin 0 -> 10256 bytes 9 files changed, 31 insertions(+), 1 deletion(-) create mode 100644 fig/spectra_eigenvalue_0.0.pdf create mode 100644 fig/spectra_eigenvalue_0.5.pdf create mode 100644 fig/spectra_eigenvalue_1.0.pdf create mode 100644 fig/spectra_eigenvalue_1.5.pdf create mode 100644 fig/spectra_singular_0.0.pdf create mode 100644 fig/spectra_singular_0.5.pdf create mode 100644 fig/spectra_singular_1.0.pdf create mode 100644 fig/spectra_singular_1.5.pdf diff --git a/bezout.tex b/bezout.tex index 880c490..eb04d2a 100644 --- a/bezout.tex +++ b/bezout.tex @@ -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 diff --git a/fig/spectra_eigenvalue_0.0.pdf b/fig/spectra_eigenvalue_0.0.pdf new file mode 100644 index 0000000..6b9b83e Binary files /dev/null and b/fig/spectra_eigenvalue_0.0.pdf differ diff --git a/fig/spectra_eigenvalue_0.5.pdf b/fig/spectra_eigenvalue_0.5.pdf new file mode 100644 index 0000000..63104d6 Binary files /dev/null and b/fig/spectra_eigenvalue_0.5.pdf differ diff --git a/fig/spectra_eigenvalue_1.0.pdf b/fig/spectra_eigenvalue_1.0.pdf new file mode 100644 index 0000000..6e3ebe1 Binary files /dev/null and b/fig/spectra_eigenvalue_1.0.pdf differ diff --git a/fig/spectra_eigenvalue_1.5.pdf b/fig/spectra_eigenvalue_1.5.pdf new file mode 100644 index 0000000..b68706e Binary files /dev/null and b/fig/spectra_eigenvalue_1.5.pdf differ diff --git a/fig/spectra_singular_0.0.pdf b/fig/spectra_singular_0.0.pdf new file mode 100644 index 0000000..86c945d Binary files /dev/null and b/fig/spectra_singular_0.0.pdf differ diff --git a/fig/spectra_singular_0.5.pdf b/fig/spectra_singular_0.5.pdf new file mode 100644 index 0000000..b25caa0 Binary files /dev/null and b/fig/spectra_singular_0.5.pdf differ diff --git a/fig/spectra_singular_1.0.pdf b/fig/spectra_singular_1.0.pdf new file mode 100644 index 0000000..4cae0c1 Binary files /dev/null and b/fig/spectra_singular_1.0.pdf differ diff --git a/fig/spectra_singular_1.5.pdf b/fig/spectra_singular_1.5.pdf new file mode 100644 index 0000000..e6f43fd Binary files /dev/null and b/fig/spectra_singular_1.5.pdf differ -- cgit v1.2.3-70-g09d2